lVIaCKay (1999) In Proceedings of the 37th Allerton Conference
`B. J. Frey and D. J. C.
`on Communication, Control and Computing 1999, Allerton House, Illinois.
`
`Irregular Turbocodes
`
`Brendan J. Frey
`
`Computer Science, University of Waterloo
`
`Electrical and Computer Engineering, University of Illinois at Urbana
`http://www.cs.uwaterloo.ca/~frey
`
`David J. C. MacKay
`
`Department of Physics, Cavendish Laboratories
`Cambridge University
`http://wol.ra.phy.cam.ac.uk/mackay
`
`Abstract
`
`Recently, several groups have increased the coding gain of iteratively decoded
`Gallager codes (low density parity check codes) by varying the number of parity
`check equations in which each codeword bit participates.
`In regular turbocodes,
`each “systematic bit” participates in exactly 2 trellis sections. We construct ir—
`regular turbocodes with systematic bits that participate in varying numbers of
`trellis sections. These codes can be decoded by the iterative application of the
`sum—product algorithm (a low—complexity, more general form of the turbodecoding
`algorithm). By making the original rate 1/2 turbocode of Berrou et al.
`slightly
`irregular, we obtain a coding gain of 0.15 dB at a block length of N = 131,072,
`bringing the irregular turbocode within 0.3 dB of capacity. Just like regular tur-
`bocodes, irregular turbocodes are linear—time encodable.
`
`1
`
`Introduction
`
`Recent work on irregular Gallager codes (low density parity check codes) has shown that
`by making the codeword bits participate in varying numbers of parity check equations,
`significant coding gains can be achieved [1—3]. Although Gallager codes have been shown
`to perform better than turbocodes at BERs below 10’5 HP, until recently Gallager codes
`performed over 0.5 dB worse than turbocodes for BERs greater than 10’5. However,
`in [3], Richardson et al. found irregular Gallager codes that perform 0.16 dB better than
`the original turbocode at BERs greater than 10—5 [5] for a block length of N % 131,072.
`
`1Gallager codes to not exhibit decoding errors, only decoding failures, at long block lengths with
`N > 5,000.
`
`Apple 1202
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`

`Permuter
`
`Permuter
`
`Permuter
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`Permuter
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`Permuter
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`Permuter
`
`Permuter
`
`

`

`Convolutional code
`
`Permuter
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`
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`f1
`
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`
`fD
`
`

`

`0.06
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`
`0.1
`
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`
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`10
`5
`0
`Degree of elite bits making up 5% of the codeword bits
`
`

`

`0.1
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`
`Figure 4: Performance of the original block length N = 131, 072 turbocode (dashed line)
`and one of its irregular cousins (solid line). These results are for irregular turbocodes
`obtained by tweaking the original turbocode — we are currently searching for optimal
`degree profiles, permuters and trellis polynomials.
`
`experiment, we simulated enough blocks to obtain a relatively small confidence interval.
`The results are shown in Fig. 3a, which indicates that for degree 10 elite bits, the best
`fraction is roughly 0.05. Next, we kept the fraction of elite bits fixed at f6 = 0.05 and
`we varied the degree of the elite bits. The results are shown in Fig. 3b, which indicates
`that for a fraction of 0.05, the best degree is roughly 10.
`
`These results show that for e = 10, f6 = 0.05 is a good fraction and that for f6 = 0.05,
`e = 10 is a good degree. However, values of e and f6 that give good profiles are probably
`correlated, so we are currently extending our search.
`
`5 Results
`
`Fig. 4 shows the simulated BER—Eb/NO curves for the original block length N = 131, 072
`regular turbocode (dashed line) and its irregular cousin (solid line), using profile 6 = 10,
`f. = 0.05.
`
`The irregular turbocode clearly performs better than the regular turbocode for BER
`> 104. At BER 2 104, the N = 131, 072 irregular turbocode is 0.3 dB from capacity,
`a 0.15 dB improvement over the regular turbocode.
`
`For high Eb/NO, most of the errors for the irregular turbocode were due to low—weight
`codewords. According to preliminary results, the distribution of error weights appears to
`indicate that the fiattening effect for the particular N = 131,072 irregular turbocode we
`constructed occurs at a higher BER than it does for the regular turbocode. However, the
`fiattening effect is highly sensitive to the technique used to construct the permuter (we
`drew it at random) and the design of the convolutional code (we just further punctured
`the convolutional code used in the original turbocode). We are currently experimenting
`with techniques for lowering the level of the fiattening effect (6.9., see [13]).
`
`

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