A Lower Bound on List Size for List Decoding

A q-ary error-correcting code C ⊆ {1,2,...,q}n is said to be list decodable to radius ρ with list size L if every Hamming ball of radius ρ contains at most L codewords of C. We prove that in order for a q -ary code to be list-decodable up to radius (1-1/q)(1- ε)n, we must have L = Ω(1/ ε2) . Specifically, we prove that there exists a constant cq > 0 and a function fq such that for small enough ε > 0, if C is list-decodable to radius (1-1/q)(1- ε)n with list size cq/ ε2, then C has at most fq( ε) codewords, independent of n . This result is asymptotically tight (treating q as a constant), since such codes with an exponential (in n ) number of codewords are known for list size L = O(1/ ε2). A result similar to ours is implicit in Blinovsky ( Problems of Information Transmission, 1986) for the binary (q=2) case. Our proof is simpler and works for all alphabet sizes, and provides more intuition for why the lower bound arises.Engineering and Applied Science

A Lower Bound on List Size for List Decoding

A q-ary error-correcting code C ⊆ {1,2,...,q}n is said to be list decodable to radius ρ with list size L if every Hamming ball of radius ρ contains at most L codewords of C. We prove that in order for a q -ary code to be list-decodable up to radius (1-1/q)(1- ε)n, we must have L = Ω(1/ ε2) . Specifically, we prove that there exists a constant cq > 0 and a function fq such that for small enough ε > 0, if C is list-decodable to radius (1-1/q)(1- ε)n with list size cq/ ε2, then C has at most fq( ε) codewords, independent of n . This result is asymptotically tight (treating q as a constant), since such codes with an exponential (in n ) number of codewords are known for list size L = O(1/ ε2). A result similar to ours is implicit in Blinovsky ( Problems of Information Transmission, 1986) for the binary (q=2) case. Our proof is simpler and works for all alphabet sizes, and provides more intuition for why the lower bound arises.Engineering and Applied Science

A Lower Bound on List Size for List Decoding

A q-ary error-correcting code C ⊆ {1, 2,..., q} n is said to be list decodable to radius ρ with list size L if every Hamming ball of radius ρ contains at most L codewords of C. We prove that in order for a q-ary code to be list-decodable up to radius (1 − 1/q)(1 − ε)n, we must have L = Ω(1/ε 2). Specifically, we prove that there exists a constant cq> 0 and a function fq such that for small enough ε> 0, if C is list-decodable to radius (1 − 1/q)(1 − ε)n with list size cq/ε 2, then C has at most fq(ε) codewords, independent of n. This result is asymptotically tight (treating q as a constant), since such codes with an exponential (in n) number of codewords are known for list size L = O(1/ε 2). A result similar to ours is implicit in Blinovsky [Bli1] for the binary (q = 2) case. Our proof is simpler and works for all alphabet sizes, and provides more intuition for why the lower bound arises.

Combinatorial limitations of average-radius list-decoding

We study certain combinatorial aspects of list-decoding, motivated by the exponential gap between the known upper bound (of $O(1/\gamma)$) and lower bound (of $\Omega_p(\log (1/\gamma))$) for the list-size needed to decode up to radius $p$ with rate $\gamma$ away from capacity, i.e., 1-\h(p)-\gamma (here $p\in (0,1/2)$ and $\gamma > 0$). Our main result is the following: We prove that in any binary code $C \subseteq \{0,1\}^n$ of rate 1-\h(p)-\gamma, there must exist a set $\mathcal{L} \subset C$ of $\Omega_p(1/\sqrt{\gamma})$ codewords such that the average distance of the points in $\mathcal{L}$ from their centroid is at most $pn$. In other words, there must exist $\Omega_p(1/\sqrt{\gamma})$ codewords with low "average radius." The standard notion of list-decoding corresponds to working with the maximum distance of a collection of codewords from a center instead of average distance. The average-radius form is in itself quite natural and is implied by the classical Johnson bound. The remaining results concern the standard notion of list-decoding, and help clarify the combinatorial landscape of list-decoding: 1. We give a short simple proof, over all fixed alphabets, of the above-mentioned $\Omega_p(\log (\gamma))$ lower bound. Earlier, this bound followed from a complicated, more general result of Blinovsky. 2. We show that one {\em cannot} improve the $\Omega_p(\log (1/\gamma))$ lower bound via techniques based on identifying the zero-rate regime for list decoding of constant-weight codes. 3. We show a "reverse connection" showing that constant-weight codes for list decoding imply general codes for list decoding with higher rate. 4. We give simple second moment based proofs of tight (up to constant factors) lower bounds on the list-size needed for list decoding random codes and random linear codes from errors as well as erasures.Comment: 28 pages. Extended abstract in RANDOM 201

A Lower Bound on the List-Decodability of Insdel Codes

For codes equipped with metrics such as Hamming metric, symbol pair metric or cover metric, the Johnson bound guarantees list-decodability of such codes. That is, the Johnson bound provides a lower bound on the list-decoding radius of a code in terms of its relative minimum distance $\delta$, list size $L$ and the alphabet size $q.$ For study of list-decodability of codes with insertion and deletion errors (we call such codes insdel codes), it is natural to ask the open problem whether there is also a Johnson-type bound. The problem was first investigated by Wachter-Zeh and the result was amended by Hayashi and Yasunaga where a lower bound on the list-decodability for insdel codes was derived. The main purpose of this paper is to move a step further towards solving the above open problem. In this work, we provide a new lower bound for the list-decodability of an insdel code. As a consequence, we show that unlike the Johnson bound for codes under other metrics that is tight, the bound on list-decodability of insdel codes given by Hayashi and Yasunaga is not tight. Our main idea is to show that if an insdel code with a given Levenshtein distance $d$ is not list-decodable with list size $L$, then the list decoding radius is lower bounded by a bound involving $L$ and $d$. In other words, if the list decoding radius is less than this lower bound, the code must be list-decodable with list size $L$. At the end of the paper we use such bound to provide an insdel-list-decodability bound for various well-known codes, which has not been extensively studied before

Bounds on List Decoding of Rank-Metric Codes

So far, there is no polynomial-time list decoding algorithm (beyond half the minimum distance) for Gabidulin codes. These codes can be seen as the rank-metric equivalent of Reed--Solomon codes. In this paper, we provide bounds on the list size of rank-metric codes in order to understand whether polynomial-time list decoding is possible or whether it works only with exponential time complexity. Three bounds on the list size are proven. The first one is a lower exponential bound for Gabidulin codes and shows that for these codes no polynomial-time list decoding beyond the Johnson radius exists. Second, an exponential upper bound is derived, which holds for any rank-metric code of length $n$ and minimum rank distance $d$. The third bound proves that there exists a rank-metric code over \Fqm of length $n \leq m$ such that the list size is exponential in the length for any radius greater than half the minimum rank distance. This implies that there cannot exist a polynomial upper bound depending only on $n$ and $d$ similar to the Johnson bound in Hamming metric. All three rank-metric bounds reveal significant differences to bounds for codes in Hamming metric.Comment: 10 pages, 2 figures, submitted to IEEE Transactions on Information Theory, short version presented at ISIT 201

Miscorrection probability beyond the minimum distance

The miscorrection probability of a list decoder is the probability that the decoder will have at least one non-causal codeword in its decoding sphere. Evaluating this probability is important when using a list-decoder as a conventional decoder since in that case we require the list to contain at most one codeword for most of the errors. A lower bound on the miscorrection is the main result. The key ingredient in the proof is a new combinatorial upper bound on the list-size for a general q−ary block code. This bound is tighter than the best known on large alphabets, and it is shown to be very close to the algebraic bound for Reed-Solomon codes. Finally we discuss two known upper bounds on the miscorrection probability and unify them for linear MDS codes