Ilya Dumer

Hardness of approximating the minimum distance of a linear code

We show that the minimum distance d of a linear code is not approximable to within any constant factor in random polynomial time (RP), unless nondeterministic polynomial time (NP) equals RP. We also show that the minimum distance is not approximable to within an additive error that is linear in the block length n of the code. Under the stronger assumption that NP is not contained in random quasi-polynomial time (RQP), we show that the minimum distance is not approximable to within the factor 2/sup log1-/spl epsi//(n), for any /spl epsi/>0. Our results hold for codes over any finite field, including binary codes. In the process, we show that it is hard to find approximately nearest codewords even if the number of errors exceeds the unique decoding radius d/2 by only an arbitrarily small fraction /spl epsi/d. We also prove the hardness of the nearest codeword problem for asymptotically good codes, provided the number of errors exceeds (2/3)d. Our results for the minimum distance problem strengthen (though using stronger assumptions) a previous result of Vardy (1997) who showed that the minimum distance cannot be computed exactly in deterministic polynomial time (P), unless P = NP. Our results are obtained by adapting proofs of analogous results for integer lattices due to Ajtai (1998) and Micciancio (see SIAM J. Computing, vol.30, no.6, p.2008-2035, 2001). A critical component in the adaptation is our use of linear codes that perform better than random (linear) codes.

Recursive decoding and its performance for low-rate Reed-Muller codes

Recursive decoding techniques are considered for Reed-Muller (RM) codes of growing length n and fixed order r. An algorithm is designed that has complexity of order nlogn and corrects most error patterns of weight up to n(1/2-/spl epsiv/) given that /spl epsiv/ exceeds n/sup -1/2r/. This improves the asymptotic bounds known for decoding RM codes with nonexponential complexity. To evaluate decoding capability, we develop a probabilistic technique that disintegrates decoding into a sequence of recursive steps. Although dependent, subsequent outputs can be tightly evaluated under the assumption that all preceding decodings are correct. In turn, this allows us to employ second-order analysis and find the error weights for which the decoding error probability vanishes on the entire sequence of decoding steps as the code length n grows.

Soft-decision majority decoding of Reed-Muller codes

We present a new soft-decision majority decoding algorithm for Reed-Muller codes RM(r,m). First, the reliabilities of 2/sup m/ transmitted symbols are recalculated into the reliabilities of 2/sup m-r/ parity checks that represent each information bit. In turn, information bits are obtained by the weighted majority that gives more weight to more reliable parity checks. It is proven that for long low-rate codes RM(r,m), our soft-decision algorithm outperforms its conventional hard-decision counterpart by 10 log/sub 10/(/spl pi//2)/spl ap/2 dB at any given output error probability. For fixed code rate R and m/spl rarr//spl infin/, our algorithm increases almost 2/sup r/2/ times the correcting capability of soft-decision bounded distance decoding.

List decoding of Reed-Muller codes up to the Johnson bound with almost linear complexity

A new deterministic list decoding algorithm is proposed for general Reed-Muller codes RM(s,m) of length n = 2m and distance d = 2m-epsi. Given n and d, the algorithm performs beyond the bounded distance threshold of d/2 and has a low complexity order of nmepsi-1 for any decoding radius T that is less than the Johnson bound

Soft-decision decoding of Reed-Muller codes: a simplified algorithm

Soft-decision decoding is considered for general Reed-Muller (RM) codes of length n and distance d used over a memoryless channel. A recursive decoding algorithm is designed and its decoding threshold is derived for long RM codes. The algorithm has complexity of order nlnn and corrects most error patterns of the Euclidean weight of order radicn/lnn, instead of the decoding threshold radicd/2 of the bounded distance decoding. Also, for long RM codes of fixed rate R, the new algorithm increases 4/pi times the decoding threshold of its hard-decision counterpart

Suboptimal decoding of linear codes: partition technique

General symmetric channels are introduced, and near-maximum-likelihood decoding in these channels is studied. First, we define a class of suboptimal decoding algorithms based on an incomplete search through the code trellis. It is proved that the decoding error probability of suboptimal decoding is bounded above for any q-ary code of length n and code rate r by twice the error probability of its maximum-likelihood decoding and tends to the latter as n grows. Second, we design a suboptimal trellis-like algorithm, which reduces the known decoding complexity of the order of q/sup n min (r,1-r)/ operations to that of q/sup nr(i-r)/ operations for all cyclic codes and virtually all long linear codes. We also consider the corresponding bounds for concatenated codes. An important corollary is that this suboptimal decoding can provide complexity below the lower bounds on trellis complexity at a negligible expense in terms of decoding error probability.

Fingerprinting Capacity Under the Marking Assumption

We study the maximum attainable rate or capacity of fingerprinting codes under the marking assumption. It is proved that capacity for fingerprinting against coalitions of size two and three over the binary alphabet satisfies 0.25 les C2,2 les 0.322 and 0.083 les C3,2 les 0.199 respectively. For coalitions of an arbitrary fixed size, we derive a closed-form upper bound on fingerprinting capacity in the binary case. Finally, for general alphabets, we establish upper bounds on the fingerprinting capacity involving only single-letter mutual information quantities.

On computing the weight spectrum of cyclic codes

Two deterministic algorithms of computing the weight spectra of binary cyclic codes are presented. These algorithms have the lowest known complexity for cyclic codes. For BCH codes of lengths 63 and 127, several first coefficients of the weight spectrum in number sufficient to evaluate the bounded distance decoding error probability are computed. >

Recursive error correction for general Reed-Muller codes

Reed-Muller (RM) codes of growing length n and distance d are considered over a binary symmetric channel. A recursive decoding algorithm is designed that has complexity of order n log n and corrects most error patterns of weight (d In d)/2. The presented algorithm outperforms other algorithms with nonexponential decoding complexity, which are known for RM codes. We evaluate code performance using a new probabilistic technique that disintegrates decoding into a sequence of recursive steps. This allows us to define the most error-prone information symbols and find the highest transition error probability p, which yields a vanishing output error probability on long codes.

Soft-decision decoding using punctured codes

Let a q-ary linear (n,k)-code be used over a memoryless channel. We design a soft-decision decoding algorithm that tries to locate a few most probable error patterns on a shorter length s /spl isin/ [k,n]. First, we take s cyclically consecutive positions starting from any initial point. Then we cut the subinterval of length s into two parts and examine T most plausible error patterns on either part. To obtain codewords of a punctured (s,k)-code, we try to match the syndromes of both parts. Finally, the designed codewords of an (s,k)-code are re-encoded to find the most probable codeword on the full length n. For any long linear code, the decoding error probability of this algorithm can be made arbitrarily close to the probability of its maximum-likelihood (ML) decoding given sufficiently large T. By optimizing s, we prove that this near-ML decoding can be achieved by using only T/spl ap/q/sup (n-k)k/(n+k)/ error patterns. For most long linear codes, this optimization also gives about T re-encoded codewords. As a result, we obtain the lowest complexity order of q/sup (n-k)k/(n+k)/ known to date for near-ML decoding. For codes of rate 1/2, the new bound grows as a cubic root of the general trellis complexity q/sup min{n-k,k}/. For short blocks of length 63, the algorithm reduces the complexity of the trellis design by a few decimal orders.