Minimal supports in linear codes

A family of error-correcting codes is list-decodable from error fraction $p$ if, for every code in the family, the number of codewords in any Hamming ball of fractional radius $p$ is less than some integer $L$ that is independent of the code length. It is said to be list-recoverable for input $\ell$ if for every sufficiently large subset of codewords (of $L$ or more), there is a coordinate where the codewords take more than $\ell$ values. The parameter $L$ is said to be the list size in either case. The capacity, i.e., the largest possible rate for these notions as the $L \to \infty$, is known to be $1-h_q(p)$ for list-decoding, and $1-\log_q \ell$ for list-recovery, where $q$ is the alphabet of the code family. In this work, we study the of random linear codes for both list-decoding and list-recovery as the rate approaches capacity. We show the following claims hold with high probability over the choice of the code (below, $\epsilon > 0$ is the gap to capacity). (1) A random linear code of rate $1 - \log_q(\ell) - \epsilon$ requires $L \ge \ell^{\Omega(1/\epsilon)}$ for list-recovery from input $\ell$. This is surprisingly in contrast to completely random codes, where $L = O(\ell/\epsilon)$ suffices w.h.p. (2) A random linear code of rate $1 - h_q(p) - \epsilon$ requires $L \ge \lfloor h_q(p)/\epsilon+0.99 \rfloor$ for list-decoding from error fraction $p$, when $\epsilon$ is sufficiently small. (3) A random binary linear code of rate $1 - h_2(p) - \epsilon$ is list-decodable from average error fraction $p$ with with $L \leq \lfloor h_2(p)/\epsilon \rfloor + 2$. The second and third results together precisely pin down the sizes for binary random linear codes for both list-decoding and average-radius list-decoding to three possible values.

Decoding random linear codes is a well studied problem with many applications in complexity theory and cryptography. The security of almost all coding and LPN/LWE-based schemes relies on the assumption that it is hard to decode random linear codes. Recently, there has been progress in improving the running time of the best decoding algorithms for binary random codes. The ball collision technique of Bernstein, Lange and Peters lowered the complexity of Stern’s information set decoding algorithm to 20.0556n. Using representations this bound was improved to 20.0537n by May, Meurer and Thomae. We show how to further increase the number of representations and propose a new information set decoding algorithm with running time 20.0494n.KeywordsInformation Set DecodingRepresentation Technique

Fast decoding algorithms for short codes based on modifications of maximum likelihood decoding algorithms of first order Reed-Muller codes are described. Only additions-subtractions, comparisons and absolute value calculations are used in the algorithms. Soft and hard decisions maximum likelihood decoding algorithms for first order Reed-Muller and the Nordstrom-Robinson codes with low complexity are proposed.

The problem of analyzing the finite-length scaling behavior of sparse random linear codes is considered. Random linear codes with random generator matrices whose entries are picked according to i.i.d. Bernoulli distribution with parameter q = o(1) are called sparse. The parameter q is referred to as the sparsity of the random linear code. We develop a methodology to show the optimality of the scaling exponent of uniform random linear codes, i.e., q = 1/2, with high probability. The results are then extended to sparse random linear codes with sparsity q = Θ(n−1/2), where n is the code block length. The encoding complexity of such sparse random linear codes is reduced from O(n2), in uniform random linear codes, to O(n 3/2). It is also conjectured that q = log n/n is the lowest sparsity of random linear codes with optimal scaling exponent. The connection of these results to an open problem regarding finding binary polar codes with optimal scaling exponent are also discussed. In particular, we point out that as the size of the polarization kernel increases it can be used as the generator matrix for a code with optimal scaling exponent, without the need to do further polarization.

It is well known that a random q-ary code of rate Ω(e2) is list decodable up to radius (1 - 1/q - e) with list sizes on the order of 1/e2, with probability 1 - o(1). However, until recently, a similar statement about random linear codes has until remained elusive. In a recent paper, Cheraghchi, Guruswami, and Velingker show a connection between list decodability of random linear codes and the Restricted Isometry Property from compressed sensing, and use this connection to prove that a random linear code of rate Ω( e2 /log3(1/e)) achieves the list decoding properties above, with constant probability. We improve on their result to show that in fact we may take the rate to be Ω(e2), which is optimal, and further that the success probability is 1 - o(1), rather than constant. As an added benefit, our proof is relatively simple. Finally, we extend our methods to more general ensembles of linear codes. As an example, we show that randomly punctured Reed-Muller codes have the same list decoding properties as the original codes, even when the rate is improved to a constant.

A random linear code has good minimal distance with high probability. The conjectured intractability of decoding random linear codes has recently found many applications in cryptography. One disadvantage of random linear codes is that their encoding complexity grows quadratically with the message length. Motivated by this disadvantage, we present a randomized construction of linear error-correcting codes which can be encoded in linear time and yet enjoy several useful features of random linear codes. Our construction is based on a linear-time computable hash function due to Ishai, Kushilevitz, Ostrovsky and Sahai [25].

Linear network coding is a promising technology that can maximize the throughput capacity of communication network. Despite this salient feature, there are still many challenges to be addressed, and security is clearly one of the most important challenges. In this paper, we will address the design of secure linear network coding. Specifically, we will investigate the network coding design that can both satisfy the weakly secure requirements and maximize the transmission data rate of multiple unicast streams between the same source and destination pair, which has not been addressed in the literature. In our study, we first prove that the secure unicast routing problem is equivalent to a constrained link-disjoint path problem. We then develop efficient algorithm that can find the optimal unicast topology in a polynomial amount of time. Based on the topology, we design deterministic linear network code that is weakly secure and can be constructed at the source node. And finally, we investigate the potential of random linear code for weakly secure unicast and prove the low bound of the probability that a random linear code is weakly secure.

The main focus in this thesis is linear codes over rings. In the first part, we look at linear codes over Galois rings, and using the homogeneous weight, we improve upon Wilson's results about the prime power that divides the coefficients of the homogeneous weight enumerators of these codes. We also prove that our results are best possible. Our results about homogeneous weight enumerators of linear codes over Galois rings generalize the results that we have for the Lee weight enumerators of linear codes over the ring of integers modulo 4. We also consider other weight enumerators, in particular the complete weight enumerators of linear codes and we obtain MacWilliams-like identities for these weight enumerators considering different rings. These MacWilliams-like identities lead to MacWilliams identities for the Hamming weight enumerators of linear codes over rings. We also give a counter-example to show that we cannot have MacWilliams-like identities for the Euclidean weight enumerators of linear codes over the ring of integers modulo 4. We also look at Gray maps from Galois rings to the finite field of prime size. We first give an application of the distance preserving Gray maps to obtain good ternary codes, and then we give an inductive algebraic construction of a distance preserving Gray map from Galois rings with the homogeneous distance to the field of prime size with the Hamming distance as well as a combinatorial construction. By using the Gray maps, we obtain some results about the weight enumerators of linear codes over the ring of dual numbers. In the last part of this work, we consider the permutation invariance of binary codes and the connection with linearity over certain rings. Among the rings considered, we have different variations and extensions of the ring of dual numbers. In this context we consider the Reed-Muller codes and answer questions about permutation invariance of Reed-Muller codes under certain permutation groups and the linearity of Reed-Muller codes over these aforementioned rings.

We introduce <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">alphabet-permutation (AP) codes</i>, a new family of error-correcting codes defined by iteratively applying random coordinate-wise permutations to a fixed initial word. A special case recovers random additive codes and random binary linear codes, where each permutation corresponds to an additive shift over a finite field. We show that when these permutations are drawn from a suitably “mixing” distribution, the resulting code is almost surely list-recoverable with list size proportional to the inverse of the gap to capacity. Compared to any linear code, our construction achieves exponentially smaller list sizes at the same rate. Previously, only fully random codes were known to attain such parameters, requiring exponentially many random bits and offering no structure. In contrast, AP codes are structured and require only polynomially many random bits—providing the first such construction to match the list-recovery guarantees of random codes.

<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Streaming codes</i> take a string of source symbols as input and output a string of coded symbols in real time, which eliminate the queueing delay of traditional <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">block codes</i> and are thus especially appealing for delay sensitive applications. This work studies the asymptotics of random linear streaming codes (RLSCs) in the large finite-field-size regime under the i.i.d. symbol erasure channel models. Two important scenarios are analyzed: (i) tradeoff between decoding deadline Δ and probability of error <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p_e</i> assuming infinite memory α = ∞; and (ii) tradeoff between α and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p_e</i> assuming infinite Δ = ∞. For each scenario, this work derives the corresponding asymptotic constant ρ, power β and decay rate η that satisfy <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p_e</i>(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i>) ∼ ρ<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">β</sup><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−η<i>x</i></sup>. The results of (i) and (ii) are then used to study an important code design problem: Under a given target deadline Δ, what is the memory length α needed for the error probability <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p_e</i> to be within a factor of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</i> > 1 of the best possible <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i>∗ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>e</i></sub> over α. Further analysis also suggests that regardless the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</i> value being considered, the necessary memory length is approximately 3–7% of the target deadline Δ when Δ is large, the actual percentage depending on the channel model and the coding rate. Such a prediction is consistent with existing brute-force-based evaluations.

Streaming codes encode a string of source packets and output a string of coded packets in real time, which eliminate the queueing delay of block coding and are thus especially suitable for delay-sensitive applications. This work studies random linear streaming codes (RLSCs) and i.i.d. packet erasure channels. While existing works focused on the asymptotic error-exponent analyses, this work characterizes the error rate in the finite memory length regime and the contributions include: (i) A new information-debt-based description of the error event; (ii) A matrix-based characterization of the error rate; (iii) A closed-form approximation of the error rate that is provably tight for large memory lengths; and (iv) A new Markov-chainbased analysis framework, which can be of independent research interest. Numerical results show that the approximation, i.e. (iii), closely matches the exact error rate even for small memory length (≈ 20). The results can be viewed as a sequential- coding counterpart of the finite length analysis of block coding [Polyanskiy et al. 10] under the specialized setting of RLSCs.

It is very well known that algebraic structures have valuable applications in the theory of error-correcting codes. Blake [Codes over certain rings, Inform. and Control 20 (1972) 396–404] has constructed cyclic codes over ℤm and in [Codes over integer residue rings, Inform. and Control 29 (1975), 295–300] derived parity check-matrices for these codes. In [Linear codes over finite rings, Tend. Math. Appl. Comput. 6(2) (2005) 207–217]. Andrade and Palazzo present a construction technique of cyclic, BCH, alternant, Goppa and Srivastava codes over a local finite ring B. However, in [Encoding through generalized polynomial codes, Comput. Appl. Math. 30(2) (2011) 1–18] and [Constructions of codes through semigroup ring [Formula: see text] and encoding, Comput. Math. Appl. 62 (2011) 1645–1654], Shah et al. extend this technique of constructing linear codes over a finite local ring B via monoid rings [Formula: see text], where p = 2 and k = 1, 2, respectively, instead of the polynomial ring B[X]. In this paper, we construct these codes through the monoid ring [Formula: see text], where p = 2 and k = 1, 2, 3. Moreover, we also strengthen and generalize the results of [Encoding through generalized polynomial codes, Comput. Appl. Math.30(2) (2011) 1–18] and [Constructions of codes through semigroup ring [Formula: see text]] and [Encoding, Comput. Math. Appl.62 (2011) 1645–1654] to the case of k = 3.

From a linear block code B over the Galois ring GR(4, to) with a k x n generator matrix and minimum Hamming distance d, a rate-k/n convolutional code over the ring Z4 with squared Euclidean free distance at least 2d and a non-recursive encoder with memory at most to m-1 is constructed. When the generator matrix of B is systematic, the convolutional encoder is systematic, basic, non-catastrophic and minimal. Long codes constructed in this manner are shown to satisfy a Gilbert-Varshamov bound.

It is shown that a finite ring R is a Frobenius ring if and only if $_R(R/\hbox {Rad}\, R)\cong \hbox {Soc}\, (_RR)$ . Other combinatorial characterizations of finite Frobenius rings are presented which have applications in the theory of linear codes over finite rings.

The combination of non-orthogonal multiple access (NOMA) and transmit antenna selection (TAS) techniques has recently attracted significant attention due to the low cost, low complexity, and high diversity gains. Meanwhile, random linear coding (RLC) is considered to be a promising technique for achieving high reliability and low latency in multicast communications. In this paper, we consider a downlink system with a multi-antenna base station and two multicast groups of single-antenna users, where one group can afford to be served opportunistically, while the other group consists of comparatively low-power devices with limited processing capabilities that have strict quality of service (QoS) requirements. In order to boost reliability and satisfy the QoS requirements of the multicast groups, we propose a cross-layer framework, including NOMA-based TAS at the physical layer and RLC at the application layer. In particular, two low-complexity TAS protocols for NOMA are studied in order to exploit the diversity gain and meet the QoS requirements. In addition, RLC analysis aims to facilitate heterogeneous users, such that sliding window-based sparse RLC is employed for computational restricted users, and conventional RLC is considered for others. Theoretical expressions that characterize the performance of the proposed framework are derived and verified through simulation results.