van Hca Henk Tilborg

On the inherent intractability of certain coding problems (Corresp.)

MEMBER, IEEE, AND HENK C. A. V~ TILBORG The fact that the general decoding problem for linear codes and the general problem of finding the weights of a linear code are both NP-complete is shown. This strongly suggests, but does not rigorously imply, that no algorithm for either of these problems which runs in polynomial time exists.

On quasi-cyclic codes with rate l m (Corresp.)

An integer linear programming problem and an additional divisibility condition are described such that they have a common solution if and only if there is a quasi-cyclic code with rate 1/m . A table of binary quasi-cyclic codes with dimensions seven and eight and rate 1/m for small m is included. In particular, there are binary linear codes with (length, dimension, minimum distance) =(35, 7,16), (42, 7,19), (80, 8, 37), (96, 8, 46) , and (112,8,54) .

The Leech lattice and the Golay code: bounded-distance decoding and multilevel constructions

Multilevel constructions of the binary Golay code and the Leech lattice are described. Both constructions are based upon the projection of the Golay code and the Leech lattice onto the (6,3,4) hexacode over GF(4). However, unlike the previously reported constructions, the new multilevel constructions make the three levels independent by way of using a different set of coset representatives for one of the quaternary coordinates. Based upon the multilevel structure of the Golay code and the Leech lattice, efficient bounded-distance decoding algorithms are devised. The bounded-distance decoder for the binary Golay code requires at most 431 operations. As compared to 651 operations for the best known maximum-likelihood decoder. Efficient bounded-distance decoding of the Leech lattice is achieved by means of partitioning it into four cosets of Q/sub 24/, beyond the conventional partition into two H/sub 24/ cosets. The complexity of the resulting decoder is only 953 real operations on the average and 1007 operations in the worst case, as compared to about 3600 operations for the best known in maximum-likelihood decoder. It is shown that the proposed algorithms decode correctly at least up to the guaranteed error-correction radius of the maximum-likelihood decoder. Thus, the loss in coding-gain is due primarily to an increase in the effective error-coefficient, which is calculated exactly for both algorithms. Furthermore, the performance of the Leech lattice decoder on the AWGN channel is evaluated experimentally by means of a comprehensive computer simulation. The results show a loss in coding-gain of less than 0.1 dB relative to the maximum-likelihood decoder for BER ranging from 10/sup -1/ to 10/sup -7/. >

On the complexity of minimum distance decoding of long linear codes

We suggest a decoding algorithm of q-ary linear codes, which we call supercode decoding. It ensures the error probability that approaches the error probability of minimum-distance decoding as the length of the code grows. For n/spl rarr//spl infin/ the algorithm has the maximum-likelihood performance. The asymptotic complexity of supercode decoding is exponentially smaller than the complexity of all other methods known. The algorithm develops the ideas of covering-set decoding and split syndrome decoding.

On the existence of optimum cyclic burst- correcting codes

It is shown that for each integer b \geq 1 infinitely many optimum cyclic b -burst-correcting codes exist, i.e., codes whose length n , redundancy r , and burst-correcting capability b , satisfy n = 2^{r-b+1} - 1 . Some optimum codes for b = 3, 4 , and 5 are also studied in detail.

Some new binary, quasi-cyclic codes

By means of local search techniques, five quasi-cyclic codes have been found that have a higher minimum distance than known binary linear codes. The new codes have parameters [102,17,37], [60,20,17], [84,21,27], [105,21,36], and [100,25,30]. Also, 39 other quasi-cyclic codes have been found that improve the parameter sets of previously known quasi-cyclic codes. Twenty-four of them give a new and easier description of binary linear codes with best known parameters.

An upper bound for codes in a two-access binary erasure channel (Corresp.)

A method for determining an upper bound for the size of a code for a two-access binary erasure channel is presented. For uniquely decodable codes, this bound gives a combinatorial proof of a result by Liao. Examples of the bound are given for codes with minimum distance 4.

Error-correcting codes with bounded running digital sum

A new approach for encoding any string of information bits into a sequence having bounded running digital sum is presented. The results improve previously known values of the running digital sum for the same rate. Also discussed are ways of incorporating an error-correcting capability into these codes. Some general constructions are given and tables are constructed for specific cases. >

On the covering radius of binary, linear codes meeting the Griesmer bound

Let g(k, d) = \sum_{i=0}^{k-1} \lceil d / 2^{i} \rceil . By the Griesmer bound, n \geq g(k, d) for any binary, linear [n, k, d] code. Let s = \lceil d / 2^{k-1} \rceil . Then, s can be interpreted as the maximum number of occurrences of a column in the generator matrix of any code with parameters [g(k, d), k, d] . Let \rho be the covering radius of a [g(k, d), k, d] code. It will be shown that \rho \leq d - \lceil s / 2 \rceil . Moreover, the existence of a [g(k, d), k, d] code with \rho = d - \lceil s / 2 \rceil is equivalent to the existence of a [g(k + 1, d), k + 1, d] code. For s \leq 2 , all [g(k,d),k,d] codes with \rho = d - \lceil s / 2 \rceil are described, while for s > 2 a sufficient condition for their existence is formulated.

Symbol synchronization in convolutionally coded systems (Corresp.)

Alternate symbol inversion is sometimes applied to the output of convolutional encoders to guarantee sufficient richness of symbol transition for the receiver symbol synchronizer. A bound is given for the length of the transition-free symbol stream in such systems, and those convolutional codes are characterized in which arbitrarily long transition free runs occur.