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`A.
`
`I don't know what this program actually
`
`189
`
`does.
`
`Just because it says: Repeatwaccumulate code
`
`simulator,
`
`that doesn't mean it actually is a
`
`repeat—accumulate simulator or has anything to do.
`
`I don't know who,
`
`in fact,
`
`the author is.
`
`I have no
`
`firsthand knowledge.
`
`I have absolutely no idea what
`
`this program does.
`
`Q.
`
`Fair enough.
`
`I'm actually asking you to
`
`set aside the particulars of this program and focus
`
`back on Divsalar Figure 3.
`
`A.
`
`Q.
`
`Okay.
`
`And what I'd like you to assume is that
`
`you've got
`
`the RA code encoder of Figure 3, okay?
`
`A.
`
`Q.
`
`Yes.
`
`And you receive an information block of
`
`length N into the repeater, okay?
`
`A.
`
`Q.
`
`Let's assume.
`
`And inside the repeater the block is
`
`divided into three subblocks: Block N1, block N2,
`
`
`
`block NK, okay?
`
`A.
`
`Q.
`
`Okay.
`
`And each own of these subblocks is
`
`repeated a different number of times, okay?
`
`A.
`
`Okay.
`
`Q.
`
`If those —— under those facts,
`
`the encoder
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`Apple vs. Caltech
`|PR201100701
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`Apme1140
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`Apple vs. Caltech
`IPR2017-00701
`Apple 1140
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`would be an IRA encoder, right?
`
`A.
`
`I have formed no opinion with respect to
`
`how IRA codes are defined in the actual patents.
`
`But, let's say, as a casual observer taking some
`
`very vaguely specified form of what
`
`IRA codes might
`
`be,
`
`that could be perhaps an interpretation.
`
`Q.
`
`Okay.
`
`That would be an IRA code as you've
`
`used it in your report, right?
`
`
`
`A.
`
`This would require a lot of assumptions in
`
`mappings between the two pictures.
`
`So I'm not
`
`claiming that this cannot be done. But this would
`
`require a very specific set of assumptions on how
`
`these numbers or how these pictures relate to.
`
`Q.
`
`Now, back in '99 and 2000, what group were
`
`you in of the classic code theorists versus the
`
`computer science physicists?
`
`A.
`
`In my Ph.D., most of my work related to
`
`questions of information theory.
`
`Information theory
`
`is kind of the abstract level of coding.
`
`So
`
`infermation theory sets limits of what can be done
`
`or not.
`
`And coding can be viewed as the kind of
`
`more applied practical way of how to actually
`
`accomplish these limits.
`
`My background is in RE.
`
`I
`
`was hired into Bell Labs into what was called the
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`mathematics of communications group.
`
`So this was a mix, people of somewhat a
`
`mix of backgrounds, most of them would have an ER
`
`background. But,
`
`for example,
`
`some people might
`
`have had a math —— math background as,
`
`for example,
`
`in the case of Dr. Shokrollahi, who got hired at
`
`some point and ——
`
`THE REPORTER: Wait. State that last part
`
`
`
`THE WITNESS:
`
`Some people might have had a
`
`math background, as was the case,
`
`I believe,
`
`for
`
`Dr. Shokrollahi, who,
`
`I think,
`
`I believe, got his
`
`degree in mathematics or perhaps computer science,
`
`but
`
`I think it was mathematics.
`
`BY MR. DOWD:
`
`Q.
`
`So the Luby group was —— was in the
`
`computer science and physicists group?
`
`A.
`
`No.
`
`The Luby group was squarely in the
`
`theoretical computer science and math group. There
`
`were various physics groups. David MacKay might be
`
`considered,
`
`to some degree, part of the physics
`
`group. But there were also other peOple working in
`
`physics being interested in these topics.
`
`Q.
`
`And what was —— what group would you place
`
`Divsalar in?
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`A.
`
`I would say he was working at JPL,
`
`if I'm
`
`192
`
`not mistaken,
`
`so he would be probably considered
`
`towards the standard classical coding group with EB
`
`backgrounds.
`
`THE REPORTER:
`
`If we —— I'm sorry.
`
`THE WITNESS: With EE backgrounds.
`
`Electric engineering.
`
`BY MR. DOWD:
`
`
`
`Q.
`
`Okay.
`
`So you were in the classical coding
`
`EE background group, right?
`
`A.
`
`My group was mixed.
`
`I myself, have that
`
`background. But
`
`in —— within Bell Labs,
`
`that group
`
`was mixed.
`
`Q.
`
`And Divsalar would have been in the same
`
`group as you the way that you've divided the world?
`
`A.
`
`He has this —— sorry.
`
`I would assume
`
`without knowing exactly his training that he is
`
`trained more classically with EB background.
`
`Q.
`
`But Dr. MacKay would have been in a
`
`different group, according to the way you're looking
`
`at the world, right?
`
`A.
`
`Dr. MaCKay played a special role because I
`
`believe he was either a student or —— or postdoc of
`
`Bob McEliece.
`
`He has a background in physics but
`
`had strong connections to this group in —— at
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`Caltech and at JPL.
`
`Q.
`
`But all of the groups, yourself,
`
`Dr. MacKay, Dr. Luby, were looking at irregular LDPC
`
`codes, right?
`
`A.
`
`I became aware of irregular LDPC codes via
`
`Dr. Shokrollahi when he got hired,
`
`I believe it was
`
`in '99 or perhaps late '98, whenever it was that he
`
`got hired. That's when I
`
`learned about
`
`the work of
`
`Luby and that group.
`
`Q.
`
`My question was, all of you were looking
`
`at irregular LDPC codes, correct?
`
`A.
`
`In a very specified sequence of timed
`
`events which had to do with how peOple got
`
`connected.
`
`Q.
`
`And your Richardson '99 paper,
`
`that was
`
`before Dr. Shokrollahi got hired at Bell Labs?
`
`No.
`
`
`
`So he was already there by that point?
`
`He's there or must have had visited.
`
`I
`
`don't knew if he was already permanently hired or
`
`not. But we had met him. That's how we learned
`
`about these works from Luby.
`
`Q.
`
`Okay. And that paper was in March
`
`of 1999, right?
`
`A.
`
`Which paper?
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`Q.
`
`Your Richardson '99 paper.
`
`A.
`
`I —~ I don't know the exact date.
`
`I have
`
`seen a preprint which dates April
`
`'99.
`
`Q.
`
`A.
`
`Okay.
`
`So April—ish 1999?
`
`Yes.
`
`I don't know if that was, you know,
`
`the exact
`
`inception date.
`
`It's a preprint that
`
`differs from the final 2001 version in some fairly
`
`substantial ways.
`
`Q.
`
`So at least as of April 1999, you knew
`
`about Luby, right?
`
`A.
`
`Yes.
`
`
`
`You knew about Divsalar, right?
`
`I would believe so, yes.
`
`And you also knew, obviously, about your
`
`own paper,
`
`the Richardson '99 paper?
`
`A.
`
`Q.
`
`Yes.
`
`Okay. And at that point,
`
`if we go back to
`
`the MacKay Ambleside '99 paper, you would have been
`
`aware of his work as well, right?
`
`A.
`
`I'm pretty Sure that I was not at the
`
`Ambleside conference. And I'm not sure to what
`
`degree I was aware of that paper that you showed me
`
`in exhibit —— the Ambleside paper, whatever ——
`
`whatever exhibit that was.
`
`Q.
`
`I believe it was Exhibit 15.
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`A.
`
`Q.
`
`15, perhaps.
`
`02
`
`:59:
`
`42
`
`Let's go back to Luby '97, which I think
`
`02
`
`:59:
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`44
`
`is Exhibit 9.
`
`02
`
`:59:
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`52
`
`Are you familiar with something called a
`
`02
`
`:59:
`
`57
`
`low—density generator matrix?
`
`Yes.
`
`A.
`
`Q.
`
`03
`
`:00:
`
`00
`
`03
`
`:00:
`
`02
`
`If I refer to that as an "LDGM," will that
`
`03
`
`:00:
`
`02
`
`make sense?
`
`A.
`
`Q.
`
`Yes.
`
`Low—density means that the matrix is
`
`sparse, right?
`
`A.
`
`Yes,
`
`that's correct.
`
`03
`
`:00:
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`07
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`03
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`:00:
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`03
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`03
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`14
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`
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`Q.
`
`And that means that it has relatively few
`
`13, mostly Os, right?
`
`A.
`
`Q.
`
`That is correct.
`
`It's called a generator matrix because
`
`it's used to generate check bits, right?
`
`A.
`
`Q.
`
`It's —— yes,
`
`that is correct.
`
`And the way it works is that you multiply
`
`the infermation bits by the matrix to get parity
`
`check bits, right?
`
`A.
`
`That is correct.
`
`So you multiply your
`
`information bit and —— with the matrix and whatever
`
`you get out would actually represent the code word
`
`that you're then transmitting.
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`Q.
`
`And we said earlier that that code word
`
`would be comprised of parity check bits, right?
`
`196
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`03
`
`:00:
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`55
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`03
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`58
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`A.
`
`You could consider that. Typically you
`
`03
`
`:01:
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`03
`
`would simply call it the code word.
`
`03
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`:01:
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`06
`
`Q.
`
`Let's turn to page, using the Bates pages
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`03
`
`:01:
`
`14
`
`along the
`
`bottom, Khandekar __
`
`03
`
`:01:
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`17
`
`Okay. Which exhibit are we talking about?
`
`03
`
`:01:
`
`22
`
`Oh, Exhibit 9.
`
`Do you have that?
`
`Yes.
`
`30 Bates page is 93? of the Luby '9?
`
`reference.
`
`Yes.
`
`03
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`
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`A.
`
`Q.
`
`80 on the left column there,
`
`there's a
`
`statement
`
`in the second full paragraph,
`
`the
`
`paragraph
`
`that begins:
`
`"It's a challenge."
`
`Do you have that paragraph?
`
`Yes.
`
`In there the second sentence says:
`
`"In this paper we present codes that
`
`can be encoded and decoded in linear time
`
`while providing near Optimal
`
`loss
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`protection."
`
`Do you see that there?
`
`Yes.
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`What does it mean to encode in linear
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`03
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`08
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`03
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`What
`
`they meant
`
`in that paper was that if
`
`03
`
`:02:
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`12
`
`you take the block length of the code, which
`
`sometimes is referred to as N, and if you look at a
`
`family of such codes where N is varied and might
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`take on different values,
`
`that the effort that is
`
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`30
`
`needed to do either the encoding or do the decoding
`
`03
`
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`:34
`
`would be a linear function of that parameter N.
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`03
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`:02:
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`40
`
`So perhaps if you had an input that was
`
`03
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`45
`
`only 100 long, it would take you, let's say,
`
`just
`
`03
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`47
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`100 operations,
`
`to keep it simple. But if you had
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`03
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`50
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`54
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`
`
`an input that was a thousand long,
`
`then it would
`
`take you thousands.
`
`So it would be proportional to
`
`the length of the input.
`
`0-
`
`And that's the same explanation of an
`
`encoding and decoding in linear time that you give
`
`in your report, right?
`
`Yes.
`
`Now,
`
`if we go over to the right column ——
`
`Yes.
`
`—— the last full paragraph in the right
`
`column that begins:
`
`"Our encoding."
`
`Do you have that?
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`The last —— sorry.
`
`The last on the right
`
`03
`
`:03:
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`29
`
`198
`
`"Our analytical tools"? Sorry.
`
`Q.
`
`The one above that,
`
`that is our
`
`encoding ——
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`03
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`THE REPORTER: Wait. Wait. Wait. One at
`
`03
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`
`a time. Start again.
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`THE WITNESS:
`
`The paragraph:
`
`"Our encoding and decoding algorithm."
`
`Yes,
`
`I see that paragraph.
`
`DOWD:
`
`Okay.
`
`It says:
`
`03
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`
`
`"Our encoding and decoding algorithms
`
`are almost symmetrical."
`
`Do you see that?
`
`Yes.
`
`What does it mean for the encoding and
`
`A.
`
`Q.
`
`decoding to be symmetrical?
`
`A.
`
`What
`
`they mean in this paper is that they
`
`use a similar type of operations to perform both,
`
`and that's what is meant with "almost symmetrical."
`
`Okay.
`
`In Luby the encoding is irregular,
`
`In —— you're talking about this particular
`
`paper, Luby '97?
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`Q.
`
`A.
`
`Correct.
`
`What do you mean with encoding is
`
`irregular? You mean whether the code is an
`
`irregular code?
`
`Q.
`
`Well,
`
`let me start there.
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`03
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`In Luby '97 the code is an irregular code,
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`correct?
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`It is a very particularly hierarchically
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`Okay. And that means that when you're
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`right?
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`column,
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`same paragraph, Page 937, it states:
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`"As
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`in many similar applications,
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`the
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`graph is chosen to be sparse, which
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`immediately implies that the encoding and
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`decoding algorithms are fast."
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`Do you see that?
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`Sorry, are we still on the same page?
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`A.
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`It's not quite clear to me what do you
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`mean by that.
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`Q.
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`A.
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`Q.
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`That doesn't make sense to
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`No.
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`Okay.
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`Now,
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`if we continue in the right
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`Yes.
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`On the right —— oh,
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`the next sentence.
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`"As
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`in many similar applications."
`
`Sorry,
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`I just —— just hold on a second.
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`Oh,
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`I see: Both are extremely simple
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`computing exactly —— okay.
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`in many similar applications —— "
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`Uh—huh.
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`~— the graph is chosen to be sparse,
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`encoding and decoding algorithms are —— "
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`there are few ls. many US?
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`A.
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`That is correct.
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`Q.
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`And so in Luby '97 you use a low—density
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`PLANET DEPOS
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`THE REPORTER: Wait. Wait.
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`If you're
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`going to read into the record. you have to read it
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`clearly and slowly.
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`THE WITNESS:
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`"As
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`in many similar applications.
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`the
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`graph is chosen to be sparse, which
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`immediately implies that the encoding and
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`decoding algorithms are fast."
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`BY MR. DOWD:
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`Q.
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`And that reference to “sparse," that
`
`refers to what we were talking about earlier about
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`generator matrix as a part of this code, right?
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`A.
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`Part of these codes can be interpreted as
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`a low—density generated.
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`Okay.
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`And part would be an LDPC.
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`And on encoding side, it's the LDGM,
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`correct?
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`A.
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`Both are,
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`in fact, used. Both the LDGM
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`and the LDPC.
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`The reason they still construct codes
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`So imagine that like a pyramid,
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`that the
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`though this is linear time encoding, it's not
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`actually a practical way of proceeding.
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`Q.
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`Well,
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`irrespective of that,
`
`let —— let's
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`PLANET DEPOS
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`LDPC, which sits kind of at the very end of the
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`pyramid at the top, has a size that is, at most,
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`square root of the total block length.
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`And so even though that part has a
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`decoding complexity to which is quite erratic,
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`the
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`overall effect it has, since it only have size which
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`is linear of the overall part, gives you still
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`something that's linear in the Overall block length.
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`Q.
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`A.
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`Okay.
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`But let me -- but also remark that even
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`return to the patents for a second.
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`You've no opinion that the claims of the
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`patents require linear or quadratic. right?
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`No.
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`A.
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`You have no opinion that the claims of the
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`practicable code. right?
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`No.
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`A.
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`Q.
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`Okay.
`
`So if we go back to Luby '9?, we
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`can agree that Luby '97 does disclose an irregular
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`LDGM;
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`is that correct?
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`A.
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`It enclose —— it —— it discloses a very
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`particular irregular and hierarchical LDGM/LDPC
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`combination.
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`Q.
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`Now.
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`if we turn to Page 930 —— withdrawn.
`
`If we go to Page 943,
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`the portion that
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`begins with the heading:
`
`"8."
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`A.
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`Q.
`
`943. Section 8?
`
`Yes.
`
`That first paragraph.
`
`if you could
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`just read that to yOurself for a moment and then let
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`me know when you've read it.
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`A.
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`Q.
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`Yes.
`
`I read it.
`
`Okay.
`
`The matrix MB that they're
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`describing there,
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`that is the generator matrix in
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`Luby, right?
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`A.
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`Right.
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`I believe that that's one —— it's
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`they have in their hierarchical description.
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`Q.
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`Okay.
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`So let's just focus on this one
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`The way this works is that you have a __ a
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`vector of N message bits, right?
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`Yes.
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`A.
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`Q.
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`That's another way of saying N information
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`bits, right?
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`A.
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`I —— I believe that all the —— I believe
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`that these are actually all the bits that you have
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`in the —— these are actually all the bits.
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`So this
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`is actually the code word itself in this case.
`
`Q.
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`So you've got N message bits, right, and
`
`that's multiplied by the beta N times N matrix MB?
`
`Right.
`
`And then that will produce check bits,
`
`Right.
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`And those are parity check bits, right?
`
`Yes.
`
`And because it says:
`
`"We Choose our graphs B to be sparse,
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`the resulting matrix MB is sparse."
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`PLANETDEPOS
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`That's how we know that it's a low—density
`
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`generator matrix, right?
`
`Yes.
`
`A.
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`Q.
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`03:
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`ll:
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`03:
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`And we also know that it is an irregular
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`low—density generator matrix because the paper's
`
`already told us that it's irregular, right?
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`03:
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`A.
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`Each component of that can be interpreted
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`03:
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`that the overall graph, of course, has some
`
`additional structure.
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`Q.
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`Okay. But I'm just foCusing on the LDGM
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`piece,
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`that would be an irregular LDGM?
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`A.
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`Right.
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`Each stage of the LDGM is,
`
`itself,
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`Each stage of the first part is an LDGM
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`03:
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`
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`an element.
`
`or it can be interpreted as an LDGM code.
`
`Q.
`
`Okay.
`
`Now,
`
`I was going to refer you to
`
`Paragraph 139 of your report where you talk about
`
`the cascade issue, but
`
`I think whether you need that
`
`or not.
`
`If we turn to Page 939,
`
`there was the
`
`Figure 2 there.
`
`THE WITNESS:
`
`I would just request a small
`
`bathroom break.
`
`It doesn't have to be this second,
`
`but
`
`I just need one minute. Unfortunately,
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`I drank
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`too much Coke.
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`50 could be any time whenever is
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`convenient for you.
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`MR. DOWD: Why don't we just finish this
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`205
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`one issue _-
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`THE WITNESS: Okay.
`
`MR. DOWD:
`
`—— then we can break.
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`THE WITNESS:
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`Sure.
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`BY MR. DOWD:
`
`Q.
`
`If we go to Page 939 in Figure 2 where it
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`says:
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`"The code levels"?
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`A.
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`Q.
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`939, Figure 2, yes.
`
`And that's what you're pointing to when
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`you're saying that there's a cascade of graphs,
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`right?
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`A.
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`Q.
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`Exactly.
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`And what that means is that you've got a
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`to the
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`next code, right?
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`A.
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`Q.
`
`Yes.
`
`Now,
`
`in a serial concatenated code,
`
`the
`
`output of the first code is the input
`
`to the second
`
`code, right?
`
`A.
`
`Q.
`
`Yes,
`
`that is correct.
`
`So that's also how serial c0ncatenated
`
`codes work, right?
`
`A.
`
`A standard definition of how serial
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`concatenated works,
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`that at least part of the output
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`VIDEOTAPED DEPOSITION OF RUDIGER L. URBANKE
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`CONDUCTED ON WEDNESDAY, FEBRUARY 25, 2015
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`of some code might be the input of something else.
`
`Q.
`
`And the RA code in Divsalar that we've
`
`been focused on is a type of serial concatenated
`
`code, right?
`
`A.
`
`You can interpret them as a serial code.
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`03:
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`MR. DOWD: Okay. Why don't we take the
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`THE VIDEOGRAPHER: Going off the record.
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`03:
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`The time is 3:14 p.m.
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`{Recess taken at 3:14 p.m.}
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`THE VIDEOGRAPHER: We are back on the
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`record.
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`The time is 3:20 p.m.
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`BY MR. DOWD:
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`03:
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`Q.
`
`So let's stick with Luby '97 and go to
`
`Page 3 —— I'm sorry, 938. And I'm looking at the
`
`Section 2,
`
`the codes.
`
`Do you see that there's a statement there,
`
`second sentence:
`
`"We begin by defining a code C(B}
`
`within message bits and beta end check
`
`bits by associating these bits with a
`
`bipartite graph B"?
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`Yes.
`
`What they're talking about there are —— is
`
`A.
`
`Q.
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`a Tanner graph representation, right?
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`20'?
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`A.
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`In hindsight, or in 2015, you might call
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`03:
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`this a Tanner graph representation, yes.
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`Q.
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`And the graph that they're talking about
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`is shown in Figure 1, right?
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`Q.
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`Yes, it's shown in Figure 1A,
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`I believe.
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`Right.
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`Now, Tanner graphs existed before
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`03:
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`1997, right?
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`A.
`
`So Tanner's paper was published,
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`I
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`believe, sometimes in the '80s.
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`But, you know,
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`just from my own
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`experience,
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`in the —— in our own '99 paper in April,
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`when we submitted it to be —— to the jOurnal, we
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`actually do not cite it. And I believe I was at
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`that point actually not aware of the paper.
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`It is
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`in the final 2001 published versi0n.
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`And I don't remember now who alerted me to
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`that paper. But at least in the '99 somehow April
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`time frame,
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`I must have not been aware of that
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`paper.
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`Q.
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`Okay. Well, setting aside what —— what
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`you were or were not aware of,
`
`in Luby '97 they're
`
`describing a bipartite graph that has message nodes
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`on the left and check bit nodes on the right?
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`Right?
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`A.
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`They're describing exactly the picture
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`VIDEOTAPED DEPOSITION OF RUDIGER L. URBANKE
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`that was given in terms in —— gathered during his
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`208
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`60s thesis.
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`Q.
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`And what
`
`they say is that the graph B has
`
`N left nodes and beta N right nodes corresponding to
`
`the message bits and the check bits respectively,
`
`right?
`
`A.
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`Q.
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`That is correct.
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`Now,
`
`to make the code irregular, you can
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`have two different degrees for the message nodes on
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`the left? Right?
`
`A.
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`To make it irregular what you have to do
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`is to choose, let's say, either variable or check
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`nodes and —— or both and decide that some of these
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`nodes within the same group would have different
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`
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`degrees.
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`Q.
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`A.
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`And that's what Luby '97 does, right?
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`Yes. Within the structure of these
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`cascaded or hierarchical LDGM/LDPC codes,
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`they
`
`introduce a notion of irregularity.
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`Q.
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`New,
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`in Luby '9?,
`
`the infermation bit
`
`variable nodes have different degree profiles,
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`right?
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`A.
`
`In the —— yes,
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`in this picture they have
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`different degrees.
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`So there's a certain fraction of
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`nodes that has a certain degree. And there's
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`something which is called a "degree profile" that
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`describes what fraction of the nodes has what
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`degree.
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`Q.
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`Okay.
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`So there's some fraction of
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`information bit nodes that has one degree and
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`another fraction of information bit nodes that has a
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`different degree. right?
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`A.
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`That is correct. There is a degree
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`nodes has what degree.
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`Q.
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`And what that means is that the first
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`fraction —— withdrawn.
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`24
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`THE REPORTER:
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`Sorry.
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`"That that..."
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`THE WITNESS:
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`Let me restart it.
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`THE REPORTER:
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`Thank you.
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`THE WITNESS:
`
`That what it means is that
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`What that means is that the information
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`bits of the first fraction will be repeated a
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`different number of times than the information bits
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`of the second fraction?
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`A.
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`What it means is that the degrees.
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`the
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`edges —— the number of edges that one such variable
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`node would have is that that. call it degree. would
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`vary ——
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`for different variables the number of edges that
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`such a variable would have would depend on to which
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`group this bit belongs to.
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`So there might be some
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`fraction of bits that perhaps has two edges
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`outgoing,
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`there's some edges that perhaps has
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`bit —— four edges going out, and there's some
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`that perhaps has five edges going out.
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`BY MR. DOWD:
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`Q.
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`Okay. And the number of edges that are
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`going out from an information node,
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`that determines
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`
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`right?
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`A.
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`"Repeat," unless you give me an exact
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`definition, which I don't think is in this paper
`
`here,
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`it simply means that in a graph the number of
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`edges that go out
`
`from such a bit is different.
`
`That's what it means.
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`Q.
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`So you don't know what "repeat" means?
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`MR. GLASS: Objection. Mischaracterizes
`
`the testimony.
`
`THE WITNESS:
`
`"Repeat" can have many, many
`
`different meanings.
`
`I don't see, yOu know,
`
`in this
`
`paper that the word "repeat" is being used, being
`
`used as an edge degree profile or as a variable node
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`degree profile.
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`So I would like to characterize it in
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`exactly the same way as they do it.
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`BY MR. DOWD:
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`Q.
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`Well, yes, no, or I don't know;
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`the degree
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`profile of an information node in an irregular LDPC
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`code corresponds to the number of repeats of the
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`information bit that will occur?
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`A.
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`Unless you give me an exact definition of
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`what "repeat" means,
`
`I cannot answer that question.
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`Q.
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`You're aware that the word "repeat" has
`
`been construed in this case?
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`A.
`
`I have no legal opinion to what's in
`
`I don't know.
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`A.
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`Q.
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`A.
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`What does "duplicate“ mean?
`
`It means create a copy of.
`
`Does an irregular LDPC code repeat bits?
`
`"Copy of," meaning in exactly what way?
`
`What do you mean with making a copy?
`
`Q.
`
`A.
`
`I mean create duplicate bits.
`
`No,
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`it simply means that there's a node
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`and the value of this node is stored somewhere, and
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`there's some certain edges going out. And these
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`respect to patents,
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`Q.
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`Okay. Let's for the sake of this question
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`assume that "repeat" means "duplicate," okay?
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`Do
`
`you have that in mind?
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`212
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`edges, you know, go to the check nodes. And this
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`number of edges that go from a particular variable
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`node differs from variable node to variable node.
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`That's what it means to me.
`
`Q.
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`Okay.
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`So I can implement an irregular
`
`code in the sense that there is a different number
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`of edges from the information node to the check
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`node? Are you with me so far?
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`A.
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`I don't think that this paper talks about
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`the implementation of how this is done.
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`It simply
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`graph in which nodes have different degrees. That's
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`
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`what
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`the paper talks about.
`
`Q.
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`Okay. Let's set this paper aside for one
`
`seCOnd and just talk about ——
`
`{Overlapping speakers.)
`
`THE REPORTER: Wait. Wait. One at a --
`
`hold on.
`
`I didn't —— you guys overlapped,
`
`so can I
`
`get a clean question, please.
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`BY MR. DOWD:
`
`Let's set the paper aside for one second,
`
`My expertise and my particular question
`
`was regarding this paper and was not about any
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`hypothetical
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`implementation.
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`213
`
`Q.
`
`I'm asking you,
`
`just set the paper aside
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`03
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`for one second; are you capable of doing that?
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`A.
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`Q.
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`Sure.
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`And I'd like you to have in mind an
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`irregular graph where the number of edges from one
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`fraction of information nodes is different than the
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`number of edges from another fraction of information
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`nodes.
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`A.
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`Q.
`
`Do you have that?
`
`Sure.
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`Okay.
`
`Now,
`
`that could be implemented
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`03
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`without repeating any of the information bits,
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`
`
`right?
`
`A.
`
`I —— you know,
`
`this paper doesn't talk
`
`about
`
`implementation.
`
`I have not
`
`thought about
`
`in
`
`this context, about how exactly such a code would be
`
`implemented.
`
`That was not
`
`the question posed to me.
`
`Q.
`
`Irrespe