Classification of Δ-Divisible Linear Codes Spanned by Codewords of Weight Δ

The binary code spanned by the rows of the point by block incidence matrix of a Steiner triple system STS(v) is studied. A sufficient condition for such a code to contain a unique equivalence class of STS(v)'s of maximal rank within the code is proved. The code of the classical Steiner triple system defined by the lines in PG(n-1,2) ( n\geq 3), or AG(n,3) ( n\geq 3) is shown to contain exactly v codewords of weight r = (v-1)/2, hence the system is characterized by its code. In addition, the code of the projective STS(2^{n}-1) is characterized as the unique (up to equivalence) binary linear code with the given parameters and weight distribution. In general, the number of STS(v)'s contained in the code depends on the geometry of the codewords of weight r. It is demonstrated that the ovals and hyperovals of the definingSTS(v) play a crucial role in this geometry. This relation is utilized for the construction of some infinite classes of Steiner triple systems without ovals.

A binary code C ⊆ F2n with minimum distance at least d and codewords of Hamming weight w is called an (n,d,w) constant weight code. The maximum size of an (n,d,w) constant weight code is denoted by A(n,d,w), and codes of this size are said to be optimal. In a computer-aided approach, optimal (n,d,w) constant weight codes are here classified up to equivalence for d=4, n ≤ 12; d=6, n ≤ 14; d=8, n ≤ 17; d=10, n ≤ 20 (with one exception); d=12, n ≤ 23; d=14, n ≤ 26; d=16, n ≤ 28; and d=18, n ≤ 28. Moreover, several new upper bounds on A(n,d,w) are obtained, leading among other things to the exact values A(12,4,5)=80, A(15,6,7)=69, A(18,8,7)=33, A(19,8,7)=52, A(19,8,8)=78, and A(20,8,8)=130 . Since A(15,6,6)=70, this gives the first known example of parameters for which A(n,d,w-1) > A(n,d,w) with w ≤ n/2. A scheme based on double counting is developed for validating the classification results.

In this paper we construct new subcodes of the second-order binary Reed-Muller code by using the permutation group and by projecting the code onto codes with smaller parameters. The permutation group of Reed-Muller codes is the general affine group and can be decomposed into the semi-direct product of the translation group and the general linear group. The action of the translation group projects the second order Reed-Muller code onto copies of the first order Reed-Muller code. The general linear group projects the code onto codes for which we can control the useful length and the dimension. These parameters depend on the dimension of the eigenspace of the chosen element of the general linear group for the eigenvalue 1.

The nonexistence of code words of weight 16 in a projective plane of order 10

Belov, Logachev and Sandimirov construct linear codes of minimum distance d for roughly 1/q k/2 of the values of d < q k-1. In this article we shall prove that, for q = p prime and roughly $${\frac{3}{8}}$$ -th's of the values of d < q k-1, there is no linear code meeting the Griesmer bound. This result uses Blokhuis' theorem on the size of a t-fold blocking set in PG(2, p), p prime, which we generalise to higher dimensions. We also give more general lower bounds on the size of a t-fold blocking set in PG(?, q), for arbitrary q and ? ? 3. It is known that from a linear code of dimension k with minimum distance d < q k-1 that meets the Griesmer bound one can construct a t-fold blocking set of PG(k?1, q). Here, we calculate explicit formulas relating t and d. Finally we show, using the generalised version of Blokhuis' theorem, that nearly all linear codes over $${{\mathbb F}_p}$$ of dimension k with minimum distance d < q k-1, which meet the Griesmer bound, have codewords of weight at least d + p in subcodes, which contain codewords satisfying certain hypotheses on their supports.

It has been well-known that the class of quasi-cyclic (QC) codes contain many good codes. In thispaper, a method to conduct a computer search for binary 2-generator QC codes is presented, anda large number of good 2-generator QC codes have been obtained. 5 new binary QC codes thatimprove the lower bounds on minimum distance are presented. Furthermore, with new 2-generatorQC codes and Construction X, 2 new improved binary linear codes are obtained. With the standardconstruction techniques, another 16 new binary linear codes that improve the lower bound on theminimum distance have also been obtained.

In this paper, we have constructed some new codes from [Formula: see text]-Simplex code called unit [Formula: see text]-Simplex code. In particular, we find the parameters of these codes and have proved that it is a [Formula: see text] [Formula: see text]-linear code, where [Formula: see text] and [Formula: see text] is a smallest prime divisor of [Formula: see text]. When rank [Formula: see text] and [Formula: see text] is a prime power, we have given the weight distribution of unit [Formula: see text]-Simplex code. For the rank [Formula: see text] we obtain the partial weight distribution of unit [Formula: see text]-Simplex code when [Formula: see text] is a prime power. Further, we derive the weight distribution of unit [Formula: see text]-Simplex code for the rank [Formula: see text] [Formula: see text].

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.

We study certain projections of binary linear codes onto larger fields. These projections include the well-known projection of the extended Golay [24,12,8] code onto the hexacode over $\mbox{GF}(4)$ and the projection of the Reed--Muller code R(2,5) onto the unique self-dual [8,4,4] code over $\mbox{GF}(4)$. We give a characterization of these projections, and we construct several binary linear codes which have best known optimal parameters, for instance, [20,11,5], [40,22,8], [48,21,12], and [72,31,16]. We also relate the automorphism group of a quaternary code to that of the corresponding binary code.

The probability of correct decoding on the binary-symmetric channel is studied. In particular, a class of codes with the same lengths and dimensions as the linear simplex codes, but with larger probability of correct decoding for all parameters p, 0 < p < 1/2, is given.

We begin with general properties of minimal supports. Then we evaluate the parameters of the distribution of the number of minimal supports in random linear codes, s tudy their asymptot ic behaviour, and discuss its consequences for some max imum likelihood decoding algorithms. In the second part , we construct explicitly sets of minimal supports in Hamming codes and second order Reed-Muller codes. Next we propose a generalization of minimal supports for codes over finite commutat ive rings and evaluate them for the quaternary Kerdock code and quaternary Hamming code. One application of minimal supports is in cryptography, namely, in the socalled linear secret-sharing schemes [1]. We briefly outline this connection and present an analog of the main result of [1] for codes over rings. Since minimal supports in linear codes correspond to cycles in representable matroids, our results have a clear interpretation in the frame of matroid theory. Details and proofs can be found in [2, 3].

Recently introduced Polar coding is the first practical coding technique that can be proven to achieve the Shannon capacity for a multitude of communication channels. Polar codes are close to Reed-Muller codes except the fact that they are tuned for the parameters of the channel. Hence, Polar codes are shown to offer better performance, e.g., in the erasure channel. It is known that second order Reed-Muller codes can be used for Compressed Sensing. Inspired by that result, we propose Polar codes as measurement matrices in CS and compare their numerical performances. We also present the algebraic relation between the erasure channel and CS theory, and discuss fast solution techniques.

Small weight codewords of projective geometric codes

It has been well-known that the class of quasi-cyclic (QC) codes contain many good codes. In this paper, a method to conduct a computer search for binary $2$-generator QC codes is presented, and a large number of good $2$-generator QC codes have been obtained. $5$ new binary QC codes that improve the lower bounds on minimum distance are presented. Furthermore, with new $2$-generator QC codes and Construction X, $2$ new improved binary linear codes are obtained. With the standard construction techniques, another $16$ new binary linear codes that improve the lower bound on the minimum distance have also been obtained.

This paper investigates the structure and complexity of low-weight subtrellises for binary linear block codes. The state and branch complexities of low-weight subtrellises for Reed-Muller (RM) codes and some extended permuted BCH (EBCH) codes are given.