`
`CONDUCTED ON WEDNESDAY, FEBRUARY 25, 2015
`
`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,
`
`Q.
`
`that could be perhaps an interpretation.
`
`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
`
`information theory sets limits of what can be done
`
`or not.
`
`And coding can be viewed as the k5
`
`
`
`more applied practical way of how to actually
`
`
`
`accomplish these limits. My background is in flfi.
`
`
`
`was hired into 2e 1 Labs into what was called
`
`
`
`
`
`
`
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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 Lfl
`
`background. But,
`
`for example,
`
`some people might
`
`
`
`have had a ma:h —— 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
`
`
`
`
`lHE WuiNLSSZ
`
`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.
`
`02:
`
`02:
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`02:
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`i 02:
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`E 02:
`é 02:
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`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
`
`
`
`02:
`
`02:
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`55:
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`O7
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`g 02:
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`O9
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`02:
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`l3
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`02:
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`55:
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`l6
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`02:
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`55:
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`21
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`02:
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`55
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`:28
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`02:
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`55:
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`31
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`02:
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`55
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`:34
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`E 02:
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`i 02:
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`37
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`55
`
`:39
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`55:
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`42
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`considered,
`
`:0 some degree, part of the physics
`
`
`group. But :here 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 99
`
`backgrounds.
`
`
`
`
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`
`
`THE REPORLLR:
`if we —~ I'm sorry.
`
`
`
`TH? WITNQSS: With E: backgrounds.
`
`
`
`
`Electric engineering.
`
`
`BY MR. DOWD:
`
`Q.
`
`Okay.
`
`So you were in the classical coding
`
`
`
`E3 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 e— sorr .
`
`I would assume
`
`
`
`
`
`without knowing exactly his training that he is
`
`
`
`trained more classically with L; 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 know 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 Aprileish 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?
`
`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.
`
`
`
`
`" be'ieve it was Exhibit 15.
`
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`A.
`
`15, perhaps.
`
`Q.
`
`Let's go back to Luby '97, which I
`
`think
`
`is Exhibit 9.
`
`Are you familiar with something called a
`
`low~density generator matrix?
`
`A.
`
`Q.
`
`Yes.
`
`If I refer to that as an "LDGM," will that
`
`
`
`make sense?
`
`A.
`
`Q.
`
`Yes.
`
`Low~density means that the matrix is
`
`sparse, right?
`
`A.
`
`Q.
`
`Yes,
`
`that's correct.
`
`And that means that it has relatively few
`
`ls, 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 information 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
`
`196 ,7
`
`would be comprised of parity check bits, right?
`
`A.
`
`You could consider that. Typically you
`
`would simply call it the code word.
`
`Q.
`
`Let's turn to page, using the Bates pages
`
`along the bottom, Khandekar “—
`
`Okay. Which exhibit are we talking about?
`
`
`Oh, Exhibit 9.
`
`Do you have that?
`
`
`Bates page is 937 of the Luby '97
`
`Yes.
`
`reference.
`
`A.
`
`Q.
`
`statement
`
`
`
`
`
`
`
`
`
`
`
`So on the left co;umn there,
`
`in the second full paragraph,
`
`
`
`there's a
`
`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
`
`protection."
`
`Do you see that there?
`
`Yes.
`
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`What does it mean to encode in linear
`
`What
`
`they meant
`
`in that paper was that if
`
`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
`
`
`take on different values,
`
`that the effort that is
`
`needed to do either the encoding or do the decoding
`
`would be a linear function of that parameter N.
`
`So perhaps if you had an input that was
`
`
`
`only 100 long, it would take you, let's say,
`
`just
`
`
`
`
`
`
`
`
`
`
`
`lOO operations,
`
`to keep it simple. But if you had
`
`
`
`
`
`an input that was a thousand long,
`
`then i: would
`
`take you thousands.
`
`So it would be propo:tional to
`
`the length of the input.
`
`Q.
`
`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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`\HDEOTAPED[EWOSHIONCHWUHMGEthURBANKE
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`CONDUCTEDCHJWEDNESDAXZFEBRUARYZiZOU
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`The last —— sorry.
`
`The last on the right
`
`198 I
`
`o3
`
`:03
`
`:29
`
`03
`
`:03:
`
`32
`
`"Our analytical tools"? Sorry.
`
`The one above that,
`
`that is our
`
`: 03:
`‘4}: 03
`
`03
`
`:32
`
`:03
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`:36
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`encoding ~—
`
`
`
`
`TH; RLPORinZ Wait. Wait. Wait.
`
`‘3 03
`
`:03:
`
`36
`
`One at
`
`o3:
`
`03:
`
`36
`
`a time. Start again.
`
` THE WITNESS:
`
`The paragraph:
`
`"Our encoding and decoding algorithm."
`
`Yes,
`
`I see that paragraph.
`
`11
`
`12
`
`13
`
`DOWD:
`
`Okay.
`
`It says:
`
`"Our encoding and decoding algorithms
`
`03
`
`:03:
`
`41
`
`03:
`
`03:
`
`41
`
`03
`
`:03:
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`42
`
`o3:
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`03:
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`45
`
`03
`
`:03:
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`47
`
`03:
`
`47
`
`03:
`
`48
`
`In we you're talking about this particular
`
`are almost symmetrical."
`
`Do you see that?
`
`Yes.
`
`What does it mean for the encoding and
`
`A.
`
`Q.
`
`E 03:
`E 03:
`i 03:
`
`g 03:
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`03:
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`50
`
`03:
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`52
`
`03
`
`:03:
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`53
`
`i 03
`3.; 03:
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`:03:
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`53
`
`03:
`
`56
`
`03:
`
`03:
`
`58
`
`E 03:
`
`04:
`
`03
`
`03:
`
`04:
`
`10
`
`f 03:
`i 03:
`g 03:
`g 03:
`
`04:
`
`13
`
`04:
`
`27
`
`04
`
`:27
`
`04
`
`:30
`
`14
`
`
`
`19
`
`20
`
`21
`
`22
`
`23
`
`24
`
`25
`
`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,
`
`paper, Luby '97?
`
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`
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`
`Q.
`
`Correct.
`
`A.
`
`What do you mean with encoding is
`
`irregular? You m an wh th r th cod
`
`is an
`
`
`
`
`
`
`
`irregular code?
`
`Q.
`
`Well,
`
`let me start there.
`
`In Luby '97 the code is an irregular code,
`
`correct?
`
`A.
`
`It is a very particularly hierarchically
`
`structured code in which some of the nodes have
`
`irregular degrees, yes.
`
`Q.
`
`Okay. And that means that when you're
`
`performing an encoding it's an irregular encoding,
`
`right?
`
`A.
`
`It's not quite clear to me what do you
`
`mean by that.
`
`That doesn't make sense to you?
`
`No.
`
`
`
`Okay.
`
`Now,
`
`if we continue in the right
`
`column,
`
`same paragraph, Page 937, it states:
`
`"As in many similar applications,
`
`the
`
`graph is chosen to be sparse, which
`
`immediately implies that the encoding and
`
`decoding algorithms are fast."
`
`Do you see that?
`
`Sorry, are we still on the same page?
`
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`Yes.
`
`On the right —— oh,
`
`the next sentence.
`
`"As
`
`in man
`
`Y
`
`similar a
`
`lications."
`
`Sorry,
`
`
`I
`
`just —— just hold on a second.
`
`Oh,
`
`I see: Both are extremely simple
`
`computing
`
`exactly —— okay.
`
`"As
`
`in many similar applications ~—
`
`u
`
`Uh-huh.
`
`”
`
`-— the graph is chosen to be sparse,
`
`which immediately implies that the
`
`
`
`encoding and decoding algorithms are w— "
`
`
`
`1H; RiPORlsR: Wait. Wait.
`
`If you're
`
`
`
`
`
`
`
`
`
`going to read into tie record, you have to read it
`
`clearly and slowly.
`
`
`THE WITN SS:
`
`
`
`"As
`
`in many similar applications,
`
`the
`
`graph is chosen to be sparse, which
`
`immediately implies that the encoding and
`
`decoding algorithms are fast."
`
`
`BY MR. DOWD:
`
`Q.
`
`And that reference to "sparse," that
`
`refers to
`
`what we were talking about earlier about
`
`there are
`
`few is, many Os?
`
`A.
`
`That is correct.
`
`Q.
`
`And so in Luby '97 you use a low~density
`
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`
`generator matrix as a part of this code, right?
`
`A.
`
`Part of these codes can be interpreted as
`
`a low~density generated.
`
`Okay.
`
`And part would be an LDPC.
`
`And on encoding side, it's the LDGM,
`
`correct?
`
`A.
`
`Both are,
`
`in fact, used. Both the LDGM
`
`and the LDPC.
`
`The reason they still construct codes
`
`that have low or linear time encoding complexity is
`
`
`
`
`
`that they have so many layers in the hierarchical
`
`structure.
`
`So imagine that like a pyramid,
`
`that the
`
`LDPC, which sits kind 0: at the very end of the
`
`
`
`
`
`pyramid at the top, has a size that is, at most,
`
`square root of the total block length.
`
`And so even though that part has a
`
`decoding complexity to which is quite erratic,
`
`the
`
`
`overall effect it has, since it only have size which
`
`
`is linear of the overall part, gives you st‘l'
`
`something that's linear in the overall block I
`
`Q.
`
`A.
`
`Okay.
`
`
`But let me ~~ but also remark that even
`
`though this is linear time encoding, it's not
`
`
`actually a practical way 0: proceeding.
`
`Q.
`
`Well,
`
`irrespective of that,
`
`let —— let‘s
`
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`
`return to the patents for a second.
`
`You‘ve no opinion that the claims of the
`
`patents require linear or quadratic, right?
`
`A.
`
`Q.
`
`No.
`
`You have no opinion that the claims of the
`
`patents actually require that it is a commercially
`
`practicable code, right?
`
`A.
`
`Q.
`
`No.
`
`Okay.
`
`So if we go back to Luby '97, we
`
`can agree that Luby '97 does disclose an irregular
`
`LDGM;
`
`is that correct?
`
`A.
`
`It enclose —— it ~~ it discloses a very
`
`particular irregular and hierarchical LDGM/LDPC
`
`combination.
`
`Now,
`
`if we turn to Page 930 ~— withdrawn.
`
`
`
`03:
`
`08:
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`02
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`03:
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`08;
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`04
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`£03:
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`08:
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`07
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`03:
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`08:
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`ll
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`:03:
`E03:
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`08:
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`12
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`08:1-
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`
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`03
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`08:
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`03:
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`08:'
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`03:
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`08:
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`03:
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`08:
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`03:
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`08:
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`41
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`08:
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`42
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`08:
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`46
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`08:
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`51
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`08:
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`52
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` describing there,
`
`A.
`
`Q.
`
`943. Sec:ion 8?
`
`
`
`:03:
`EE03:
`E 03:
`:03:
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`o3:
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`08:
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`56
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`§03:
`:03:
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`09:
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`00
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`09:
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`04
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`03:
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`09:
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`18
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`:03:
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`03
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`09:
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`22
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`09
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`:25
`
`03:
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`10:
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`l6
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`O3
`
`l0:
`
`l8
`
`03:
`
`10:
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`22
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`10:
`
`27
`
`l7
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`l8
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`19
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`20
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`21
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`22
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`23
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`24
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`25
`
`Q.
`
`If we go to Page 943,
`
`the portion that
`
`begins with the heading:
`
`"8."
`
`Yes. Tha: first paragraph,
`
`if you could
`
`just read that to yourself for a moment and then let
`
`me know when you've read it.
`
`A.
`
`Q.
`
`Yes,
`
`I read it.
`
`Okay.
`
`The matrix MB that they're
`
`that is the genera:or matrix in
`
`Luby, right?
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`A.
`
`Right.
`
`I believe that that's one —— it's
`
`203
`
`a matrix that corresponds to one of the stages that
`
`they have in their hierarchical description.
`
`Q.
`
`Okay.
`
`So let's just focus on this one
`
`The way this works is that you have a _i a
`
`vector of N message bits, right?
`
`A.
`
`Q.
`
`Yes.
`
`That's another way of saying N information
`
`bits, right?
`
`A.
`
`
`I —— I believe that all the ~— I believe
`
`Q.
`
`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.
`
`And those are parity check bits, right?
`
`Yes.
`
`And because it says:
`
`
`"We choose our graphs 3 to be sparse,
`
`the resulting matrix MB is sparse."
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`that these are actually all the bits that you have
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`in the ~— these are actually all tie bits.
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`So this
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`
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` is actually the code word itself in this case.
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`That's how we know that it's a low‘density
`
`204
`
`generator matrix, right?
`
`A.
`
`Q.
`
`Yes.
`
`And we also know that it is an irregular
`
`low—density generator matrix because the paper's
`
`already told us that it's irregular, right?
`
`A.
`
`
`
`Each component 0: that can be interpreted
`
`that the overall graph, of course, has some
`
`additional structire.
`
`
`
`Q.
`
`Okay. But I'm just focusing on the LDGM
`
`piece,
`
`that would be an irregular LDGM?
`
`A.
`
`Right.
`
`
`Each stage of the LDGM is, itself,
`
`an element.
`
`
`Each stage of the first part is an LDGM
`
`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
`
`03.” not .
`
` If we turn to Page 939,
`
`there was the
`
`Figure 2 there.
`
`
`
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`
`
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`
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`
`
`THE WITNESS:
`
`I would just request a small
`
`bathroom break.
`
`
`I: doesn't have to be this second,
`
`
`
`but
`
`I
`
`just need one minute. Unfortunately,
`
`I drank
`
`too much Coke.
`
`So could be any time whenever is
`
`convenient for you.
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`MR. DOWD: Why don't we just finish this
`
`205
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`one issue ——
`
`
`
`
`
`lH; W lN?SS: Okay.
`
`MR. DOWD:
`
`
`
`
`lHfl WilNLSS:
`
`Sure.
`
`~— then we can break.
`
`
`
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`
`
`BY MR. DOWD:
`
`Q.
`
`If we go to Page 939 in Figure 2 where it
`
`says:
`
`"The code levels"?
`
`A.
`
`Q.
`
`939, Figure 2, yes.
`
`And that's what you're pointing to when
`
`you're saying that there's a cascade of graphs,
`
`right?
`
`A.
`
`Q.
`
`
`Exactly.
`
`And what that means is that you've got a
`
`
`graph 0: one code whose output is the input to the
`
`next code, right?
`
`A.
`
`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 concatenated
`
`codes work, right?
`
`A.
`
`A standard definition of how serial
`
`concatenated works,
`
`
`that at least part of the output
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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.
`
`MR. DOWD: Okay. Why don't we take the
`
`
`
`
`
`
`
`1H; V D?OGRAPHER: Going off the record.
`
`The time is 3:14 p.m.
`
`(Recess taken at 3:14 p.m.)
`
`
`
`THE VIDEOGRAPHER: We are back on the
`
`record.
`
`The time is 3:20 p.m.
`
`BY MR. DOWD:
`
`:00
`
`:04
`
`:05
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`:09
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`:11
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`:15
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`:17
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`:18
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`:20
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`»:22
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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 cueck
`
`bits by associating these bits with a
`
`
`bipartite graph 3"?
`
`Yes.
`
`What they're talking about there are —— is
`
`A.
`
`Q.
`
`a Tanner graph representation, right?
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`A.
`
`In hindsight, or in 2015, you might call
`
`207 f
`
`this a Tanner graph representation, yes.
`
`Q.
`
`And the graph that they're talking about
`
`is shown in Figure I, right?
`
`A.
`
`Q.
`
`Yes, it's shown in Figure 1A,
`
`
`I believe.
`
`Right.
`
`Now, Tanner graphs existed before
`
`1997, right?
`
`A.
`
`So Tanner's paper was published,
`
`I
`
`believe,
`
`sometimes in the '808.
`
`But, you know,
`
`just from my own
`
`
`
`
`
`
`
`
`
`
`
`
`
`experience,
`
`in the —— in our own '99 paper in April,
`
`when we submitted it to be —« to the journal, we
`
`actually do not cite it. And I believe I was at
`
`that point actually not aware of the paper.
`
`It is
`
`in the final 2001 published version.
`
`And I don't remember now who alerted me to
`
`that paper. But at least in the '99 somehow April
`
`time frame,
`
`I must have not been aware of that
`
`paper.
`
`Q.
`
`Okay. Well, setting aside what —~ what
`
`you were or were not aware of,
`
`in Luby '97 they're
`
`describing a bipartite graph that has message nodes
`
`on the left and check bit nodes on the right?
`
`Right?
`
`A.
`
`They‘re describing exactly the picture
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`that was given in terms in *~ gathered during his
`
`208 "
`
`60s thesis.
`
`Q.
`
`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.
`
`Q.
`
`That is correct.
`
`Now,
`
`to make the code irregular, you can
`
`
`have two different degrees for the message nodes on
`
`the left? Right?
`
`A.
`
`To make it irregular what you have to do
`
`is to choose, let's say, either variable or check
`
`nodes and —e or both and decide that some of these
`
`nodes within the same group would have different
`
`degrees.
`
`Q.
`
`A.
`
`And that's what Luby '97 does, right?
`
`Yes. Within the structure of these
`
`cascaded or hierarchical LDGM/LDPC codes,
`
`they
`
`introduce a notion of irregularity.
`
`Q.
`
`Now,
`
`in Luby '97,
`
`the information bit
`
`variable nodes have different degree profiles,
`
`right?
`
`So there's a certain fraction of
`
`
`
`
`
`A.
`
`In the ~— yes,
`
`in this picture they have
`
`different degrees.
`
`nodes that has a certain degree. And there's
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`something which is called a "degree profile" that
`
`209 __
`
`describes what fraction of the nodes has what
`
`degree.
`
`Q.
`
`Okay.
`
`So there's some fraction of
`
`information bit nodes that has one degree and
`
`another fraction of information bit nodes that has a
`
`
`
`
`
`
`different degree, right?
`
`A.
`
`That is correct.
`
`There is a degree
`
`profile that describes what fraction of the various
`
`nodes has what degree.
`
`Q.
`
`And what that means is that the first
`
`fraction —— withdrawn.
`
`What that means is that the information
`
`bits of the first fraction will be repeated a
`
`different number of times than the information bits
`
`
`of the second fraction?
`
`A.
`
`What it means is that the degrees,
`
`the
`
`edges —w the number of edges that one such variable
`
`node would have is that that, call it degree, would
`
`vary —~
`
`
`
`E REPORTER:
`
`Sorry.
`
`
`
` "Tqat that..."
`
`
`
`lHE WllNESS:
`
`
`
`
`THE REPORTER:
`
`Let me restart it.
`
`Thank you.
`
`THE WITNESS: That what it means is that
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`
`for different variables the number of edges that
`
`such a variable would have would depend on to which
`
`group this bit belongs to.
`
`So there might be some
`
`fraction of bits that perhaps has two edges
`
`outgoing,
`
`there's some edges that perhaps has four
`
`bit —— four edges going out, and there‘s some bits
`
`:hat perhaps has five edges going out.
`
` BY MR. DOWD:
`
`
`
`
`
`
`
`Q.
`
`Okay. And the number of edges that are
`
`going out
`
`from an information node,
`
`that determines
`
`how many times the bit of that node is repeated,
`
`right?
`
`A.
`
`"Repeat," unless you give me an exact
`
`definition, which I don't think is in this paper
`
`here, it simply means that in a graph the number of
`
`edges that go out
`
`from such a bit is different.
`
`That's what it means.
`
`Q.
`
`So you don't know what "repeat" means?
`
`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
`
`degree profile.
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`So I would like to characterize it in
`
`exactly the same way as they do it.
`
`
`BY MR. DOWD:
`
`Q.
`
`Well, yes, no, or I don't know;
`
`the degree
`
`profile of an information node in an irregular LDPC
`
`code corresponds to the number of repeats of the
`
`information bit that will occur?
`
`A.
`
`Unless you give me an exact definition of
`
`what "repeat" means,
`
`I cannot answer that question.
`
`Q.
`
`You're aware that the word "repeat" has
`
`been construed in this case?
`
`A.
`
`I have no legal opinion to what's in
`
`
`
`
`
`respect to patents,
`
`I don't know.
`
`Q.
`
`
`Okay. Let's for the sake of this question
`
`assume that "repeat" means "duplicate," okay?
`
`Do
`
`you have that in mind?
`
`A.
`
`Q.
`
`A.
`
`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, it simply means that there's a node
`
`and the value of this node is stored somewhere, and
`
`there's some certain edges going out. And these
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`edges, you know, go to the check nodes. And this
`
`212
`
`number of edges that go from a particular variable
`
`node differs from variable node to variable node.
`
`
`
`That's what it means to me.
`
`Q.
`
`Okay.
`
`So I can implement an irregular
`
`code in the sense that there is a different number
`
`of edges from the information node to the check
`
`node? Are you with me so far?
`
`A.
`
`I don't think that this paper talks about
`
`the implementation of how this is done.
`
`It simply
`
`talks about a mathematical concept of a bipartite
`
`graph in which nod s hav different d gr
`
`5. That's
`
`what the paper talks about.
`
`Q.
`
`Okay. Let's set this paper aside for one
`
`second and just talk about M~
`
`(Overlaoping speakers.)
`
`
`
`
`
`
`THE REPORTER: Wait. Wait.
`
`One at a ~—
`
`hold on.
`
`I dida't —— you guys overlapped,
`
`so can I
`
`get a clean question, please.
`
`
`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
`
`
`
`hypothetical implementation.
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`Q.
`
`I'm asking you,
`
`just set the paper aside
`
`213 {i
`
`for one second; are you capable of doing that?
`
`A.
`
`Q.
`
`Sure.
`
`And I‘d like you to have in mind an
`
`irregular graph where the number of edges from one
`
`
`fraction of information nodes is different than the
`
`number of edges from another fraction of information
`
`nodes.
`
`A.
`
`Q.
`
`Do you have that?
`
`Sure.
`
`Okay.
`
`Now,
`
`that could be implemented
`
`without repeating any of the information bits,
`
`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.
`
`Irrespective of the question posed to you,
`
`
`
`can you tell me the answer?
`
`
`
`I don‘t know.
`
`A.
`
`Q.
`
`Okay.
`
`So let's get back to ~— let's get
`
`back to our irregular graph.
`
`
`In the case where you
`
`have some fraction of information nodes with one
`
`number of edges, another fraction with a different
`
`number of edges,
`
`am I correct that the information
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`nodes of the first fraction will contribute to a
`
`
`different number of parity checks than the second
`
`fraction?
`
`A.
`
`Q.
`
`Not clear.
`
`All right.
`
`It could be or could not be.
`
`So at Least what we know is,
`
`
`
`
`
`
`
`
`
`when you read Luby '97, one way to make the
`
`bipartite graph irregular is that you can have one
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`fraction with one degree profile,
`
`a different
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`fraction with a different degree profile, right?
`
`A.
`
`The profile actually refers to the
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`whole ‘~ to «~ to the set of all these fractions.
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`So the profile already specifies for each set.
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`So
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`what is meant typically as a profile is simply
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`there's a certain probability or certain fraction
`
`
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`a certain fraction to
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`that applies to some set,
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`another set.
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`Q.
`
`Let me ask you a better question,
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`then.
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`What we can know from Luby ‘97 is that one
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`way to make an irregular graph is to have one
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`fraction of information nodes with one number
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`edges and a different fraction of information
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`with a different number of edges, correct?
`
`A.
`
`Q.
`
`Yes,
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`that's what the degree profile
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`All right.
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`MR. DOWD:
`
`
`Now, let's mark as Exhibit 17,
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`VIDEOTAPED DEPOSITION OF RUDIGER L. URBANKE
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`a copy of Luby '98.
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`(Urbanke Exhibit 17 was marked for
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`identification and attached to the
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`transcript.)
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`DOWD:
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`
`Do you recognize Exhibit 17?
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`Yes.
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`What is it?
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`
`
`
`
`Appears to be the ~~ what we called Luby
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`'98 paper.
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`Its title is:
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`"Analysis of Low—Density
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`Codes and Improved the Science Using Irregular
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`Graphs."
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`Q.
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`And if you could turn to Page 925.
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`In the
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`right~hand column,
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`there's a paragraph that begins:
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`"The main contribution."
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`Do you see that there?
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`Yes.
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`It says:
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`"The main contribution of this paper
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`is the design and analysis of low—density
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`parity check codes based on irregular
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`graphs. This work follows the general
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`approach introduced in 7 for the design
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`and analysis of erasure codes."
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`Do you see that?
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`Yes.
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`7 is a reference to Luby '97, right?
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`So ——
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`
`r
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`lHr
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`RIER: Did you answer?
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`
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` THE
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`THE
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`ESS: Yes.
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`E
`iR:
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`Thank you.
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`BY MR. DOWD:
`
`Q.
`
`So Luby '98 says that Luby ‘97 was a
`
`general approach to irregular codes, right?
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`
`
`
`
`
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`A.
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`I believe the way I read it that "general"
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`here doesn't mean in general is applicable to a
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`general set of channels or a general set of graphs,
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`but it means, you know,
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`the —~
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`the approach,
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`essentially you can skip the "general" here.
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`It
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`doesn't mean general
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`in the sense of applicable to a
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`general class or a general channel.
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`Q.
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`So the way you read it is you strike the
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`word "general” from the sentence?
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`MR. GLASS: Objection. Mischaracterizes
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`
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`the testimony.
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`1H; W lNVSS:
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`
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`The way I read it is ~~ is,
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`
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`you know, without being, you know, anything
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`specific.
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`So not a specific thing, but, you know,
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`an idea that was put forth in that paper. And so
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`they're saying that it shares some characteristics
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`with that idea.
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`BY MR. DOWD:
`
`Q.
`
`The next sentence of Luby '98 says:
`
`"There," referring to Luby '97, right?
`
`A.
`
`Q.
`
`A.
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`Q.
`
`Uh-huh.
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`"There it is shown that using
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`irregular graphs yields codes with much
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`better performance than regular graphs."
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`Right?
`
`Yes.
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`And there's no mention there of any
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`specific type of code, right?
`
`A.
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`That is correct. But
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`the paper
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`'97 only
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`
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`
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`
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`
`
`
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`deals with a very specific cnannel limited BC, a
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`very specific decoding a_gorithm; namely, what is
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`
`
`
`
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`message passing for the 3C, which is also called a
`
`peeling decoder, and witi a very specific code
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`structure.
`
`Q.
`
`Well,
`
`in Luby '98,
`
`the statement that:
`
`"...irregular graphs yield codes with
`
`much better performance than regular
`
`graphs,"
`
`that isn't talking about any specific
`
`code, right? There's no code named there, right?
`
`A.
`
`The only thing that is in the '97 paper is
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`a very specific code,
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`a very specific channel, and a
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`218
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`very specific decoding algorithm.
`
`So the only thing
`
`that can be claimed is what exactly is in that
`
`paper.
`
`Q.
`
`Well,
`
`I'm asking a different question.
`
`My question is,
`
`if you look at Luby '98,
`
`the statement:
`
`"...that using irregular graphs yields
`
`codes with much better performance than
`
`regular graphs."
`
`That statement is made out without naming
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`
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`
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`any specific code, correct?
`
`A.
`
`In that particular statement,
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`they don't
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`
`name any codes. But
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`they don't give any evidence
`
`that that would be true.