On Z/sub 4/-duality

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.

This paper examines the w w -weight enumerators of weights w w with maximal symmetry over finite chain rings and matrix rings over finite fields. In many cases, including the homogeneous weight, the MacWilliams identities for w w -weight enumerators fail because there exist two linear codes with the same w w -weight enumerator whose dual codes have different w w -weight enumerators.

A partial description of (complete) weight enumerators of linear self-dual codes is given. 0. Let F = Z=pZ, where p is a prime number. If C is a linear code on F of length n, i.e., a linear subspace in F n , then its (complete) weight enumerator WC is deÞned to be X u2C e Y a2F x s a(u) a e

In Chapter 6, we studied Reed–Solomon codes, codes whose codewords are the evaluation of polynomials in one variable of degree at most k − 1 at the elements of \({\mathbb F}_q \cup \{\infty \}\). Reed–Solomon codes are short length codes, where the length n is bounded by q + 1, and only useful when we take the field to be large. The alternant codes which we constructed from generalised Reed–Solomon codes in Chapter 7 allowed us to construct codes over small fields and we put this to good use. In this chapter we will consider another generalisation of Reed–Solomon codes, codes whose codewords are the evaluation of polynomials in many variables. This again allows us to construct linear codes over small fields and we will restrict our attention, for the most part, to binary linear codes. It will turn out that these codes are not asymptotically good. Nevertheless, they are an important class of codes which are widely implemented due to the availability of fast decoding algorithms. One example of such a decoding algorithm is the majority-logic decoding algorithm that we will study here. We will then go on and construct Kerdock codes which are certain subcodes of the second-order Reed–Muller codes. These codes can give examples of non-linear codes with parameters for which no linear code exists.

In this paper, we study a type of mixed alphabet codes, called [Formula: see text]-linear codes, where [Formula: see text] with [Formula: see text]. First, we define and discuss several weight enumerators of [Formula: see text]-linear codes, including Hamming weight, Lee weight and symmetrized weight enumerators. MacWilliams identity connects the weight enumerator of codes and their dual codes. Further, we present the MacWilliams identities about Lee weight and complete weight over the ring [Formula: see text]. Finally, several examples are presented to illustrate our main results.

This paper studies linear and affine error-correcting codes for correcting synchronization errors such as insertions and deletions. We call such codes linear/affine insdel codes. Linear codes that can correct even a single deletion are limited to having an information rate at most (achieved by the trivial two fold repetition code). Previously, it was (erroneously) reported that more generally no nontrivial linear codes correcting deletions exist, i.e., that the -fold repetition codes and its rate of are basically optimal for any . We disprove this and show the existence of binary linear codes of length and rate just below capable of correcting insertions and deletions. This identifies rate as a sharp threshold for recovery from deletions for linear codes and reopens the quest for a better understanding of the capabilities of linear codes for correcting insertions/deletions. We prove novel outer bounds and existential inner bounds for the rate vs. (edit) distance trade-off of linear insdel codes. We complement our existential results with an efficient synchronization-string-based transformation that converts any asymptotically good linear code for Hamming errors into an asymptotically good linear code for insdel errors. Last, we show that the -rate limitation does not hold for affine codes by giving an explicit affine code of rate which can efficiently correct a constant fraction of insdel errors.

Near MDS codes of non-elliptic-curve type from Reed-Solomon codes

Nonlinear codes defined by quadratic forms over GF(2)

A general case of normal duality between binary nonlinear codes

In [4] it was shown, that the weight enumerators of two binary ℤ4-linearly dual codes satisfy the McWilliams identity (i.e. these codes are formally dual). If we consider an arbitrary Galois ring R=GR(q 2, p 2) of characteristic p 2 and a pair of R-linearly dual codes over a Galois field GF(q) this result is not preserved. We propose the approach to correcting this disadvantage. The titled codes are presented as codes inthe alphabet ℜ=RS q (q,2), being a Reed-Solomon code. The appropriate exact weight enumerators of these codes are reduced to some projective weight enumerators (obtained by identifying of variables) which satisfy the McWilliams identity for linear codes over GF(q). We discuss ways of “optimal” identifying of variables such that the corresponding projective weight enumerators allow us to construct complete weight enumerators of the initial codes over GF(q).

A code $${{\mathcal C}}$$ is $${{\mathbb{Z}_2\mathbb{Z}_4}}$$ -additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of $${{\mathcal C}}$$ by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of $${{\mathbb{Z}_2\mathbb{Z}_4}}$$ -additive codes under an extended Gray map are called $${{\mathbb{Z}_2\mathbb{Z}_4}}$$ -linear codes. In this paper, the invariants for $${{\mathbb{Z}_2\mathbb{Z}_4}}$$ -linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of $${{\mathbb{Z}_2\mathbb{Z}_4}}$$ -linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a $${{\mathbb{Z}_2\mathbb{Z}_4}}$$ -linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a $${{\mathbb{Z}_2\mathbb{Z}_4}}$$ -linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a $${{\mathbb{Z}_2\mathbb{Z}_4}}$$ -linear code for each possible pair (r, k) is given.

A formal weight enumerator is a homogeneous polynomial in two variables which behaves like the Hamming weight enumerator of a self-dual linear code except that the coefficients are not necessarily nonnegative integers. The notion of formal weight enumerator was first introduced by Ozeki in connection with modular forms, and a systematic investigation of formal weight enumerators has been conducted by Chinen in connection with zeta functions and Riemann hypothesis for linear codes. In this paper, we establish a relation between formal weight enumerators and Chebyshev polynomials. Specifically, the condition for the existence of formal weight enumerators with prescribed parameters $$(n,\varepsilon ,q)$$ is given in terms of Chebyshev polynomials. According to the parity of n and the sign $$\varepsilon$$ , the four kinds of Chebyshev polynomials appear in the statement of the result. Further, we obtain explicit expressions of formal weight enumerators in the case where n is odd or $$\varepsilon =-1$$ using Dickson polynomials, which generalize Chebyshev polynomials. We also state a conjecture from a viewpoint of binomial moments, and see that the results in this paper partially support the conjecture.

Linear codes constitute an important family of error-correcting codes and have a rich algebraic structure. Initially, these codes were studied with respect to the Hamming metric; while for the past few years, they are also studied with respect to a non-Hamming metric, known as the Rosenbloom–Tsfasman metric (also known as RT metric or ρ metric). In this paper, we introduce and study the split ρ weight enumerator of a linear code in the R-module Mn×s(R) of all n × s matrices over R, where R is a finite Frobenius commutative ring with unity. We also define the Lee complete ρ weight enumerator of a linear code in Mn×s(ℤk), where ℤk is the ring of integers modulo k ≥ 2. We also derive the MacWilliams identities for each of these ρ weight enumerators.

Certain nonlinear binary codes contain more codewords than any comparable linear code presently known. These include the Kerdock and Preparata codes, which exist for all lengths 4^m ≥ 16. At length 16 they coincide to give the Nordstrom-Robinson code. This paper constructs a nonlinear (64, 2^37, 12) code as the binary image, under the Gray map, of an extended cyclic code defined over the integers modulo 4 using Galois rings. The Nordstrom-Robinson code is defined in this same way, and like the Nordstrom-Robinson code, the new code is better than any linear code that is presently known.

We introduce the Poincare polynomial of a linear q-ary code and its relation to the corresponding weight enumerator. The question of whether the Poincare polynomial is a complete invariant is answered affirmatively for q = 2, 3 and negatively for q ≥ 4. Finally we determine this polynomial for MDS codes and, by means of a recursive formula, for binary Reed-Muller codes.