On distances and self-dual codes over Fq[u]∕(ut)

The purpose of this paper is to study codes over finite principal ideal rings. To do this, we begin with codes over finite chain rings as a natural generalization of codes over Galois rings GR(p e , l) (including $${\mathbb{Z}_{p^e}}$$ ). We give sufficient conditions on the existence of MDS codes over finite chain rings and on the existence of self-dual codes over finite chain rings. We also construct MDS self-dual codes over Galois rings GF(2 e , l) of length n = 2 l for any a ? 1 and l ? 2. Torsion codes over residue fields of finite chain rings are introduced, and some of their properties are derived. Finally, we describe MDS codes and self-dual codes over finite principal ideal rings by examining codes over their component chain rings, via a generalized Chinese remainder theorem.

In this chapter, we describe self-dual codes over Frobenius rings. We give constructions of self-dual codes over any Frobenius ring. We describe connections to unimodular lattices, binary self-dual codes and to designs. We also describe linear complementary dual codes and make a new definition of a broad generalization encompassing both self-dual and linear complementary codes.

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 linear cyclic codes over the ring F/sub 2/+uF/sub 2/={0,1,u,u~=u+1}, where u/sup 2/=0 and study them by analogy with the Z/sub 4/ case. We give the structure of these codes on this new alphabet. Self-dual codes of odd length exist as in the case of Z/sub 4/-codes. Unlike the Z/sub 4/ case, here free codes are not interesting. Some nonfree codes give rise to optimal binary linear codes and extremal self-dual codes through a linear Gray map.

Optimal double circulant self-dual codes over F/sub 4/ have been found for each length n/spl les/40. For lengths n/spl les/14, 20, 22, 24, 28, and 30, these codes are optimal self-dual codes. For length 26, the code attains the highest known minimum weight. For n/spl ges/32, the codes presented provide the highest known minimum weights. The [36,18,12] self-dual code improves the lower bound on the highest minimum weight for a [36,18] linear code.

This paper investigates the structures and properties of one-Lee weight codes and two-Lee weight projective codes over ℤ4. The authors first give the Pless identities on the Lee weight of linear codes over ℤ4. Then the authors study the necessary conditions for linear codes to have one-Lee weight and two-Lee projective weight respectively, the construction methods of one-Lee weight and two-Lee weight projective codes over ℤ4 are also given. Finally, the authors recall the weight-preserving Gray map from (ℤ 4 , Lee weight) to ( $\mathbb{F}_2^{2n} $ , Hamming weight), and produce a family of binary optimal oneweight linear codes and a family of optimal binary two-weight projective linear codes, which reach the Plotkin bound and the Griesmer bound.

In this chapter, we shall present some fundamental study on relatively self-dual codes over a finite commutative ring. Section 1.1 is devoted to the basic definitions and properties of such codes. In Section 1.2, we shall present our gluing technique for constructing relatively self-dual codes. Several exotic general constructions of self-dual codes over ℤn will be given in Section 1.3. Four constructions of binary self-dual codes by our gluing technique will be given in Section 1.4. In Section 1.5, we shall give some constructional examples of ternary self-dual codes. In Section 1.6, we shall present some families of self-dual codes over ℤ4. Sections 1.1, 1.2 and Constructions III, IV in Section 1.4 were published in [X2].

In this work, we study codes over the ring 𝓡 k, m = 𝔽2[u, v]/ 〈uk , vm , uv − vu〉, which is a family of Frobenius, characteristic 2 extensions of the binary field. We introduce a distance and duality preserving Gray map from 𝓡 k, m to F 2 k m $\mathbb{F}_2^{km}$ together with a Lee weight. After proving the MacWilliams identities for codes over 𝓡 k, m for all the relevant weight enumerators, we construct many binary self-dual codes as the Gray images of self-dual codes over 𝓡 k, m . In addition to many extremal binary self-dual codes obtained in this way, including a new construction for the extended binary Golay code, we find 175 new Type I binary self-dual codes of parameters [72, 36, 12] and 105 new Type II binary self-dual codes of parameter [72, 36, 12].

Abstract. We present two kinds of construction methods for self-dualcodes over F 2 + u F 2 . Specially, the second construction (respectively,the rst one) preserves the types of codes, that is, the constructed codesfrom Type II (respectively, Type IV) is also Type II (respectively, TypeIV). Every Type II (respectively, Type IV) code over F 2 + u F 2 of freerank larger than three (respectively, one) can be obtained via the secondconstruction (respectively, the rst one). Using these constructions, weupdate the information on self-dual codes over F 2 + u F 2 of length 9 and 10,in terms of the highest minimum (Hamming, Lee, or Euclidean) weightand the number of inequivalent codes with the highest minimum weight. 1. IntroductionCodes over Z 4 have had great attention since the monumental paper [5],and self-dual codes over F 2 + u F 2 have also received some attention lately [1, 3]due to their importance. In fact, codes over Z 4 and F 2 + u F 2 are relatedto lattices, and they have Gray maps [3]. Even though they are similar toeach other, there are still some signi cant ff between them; one ofthe ff is that

Let R be an arbitrary commutative finite chain ring and \(\gamma \) a fixed generator of the maximal ideal of R. Suppose s is the nilpotency index of \(\gamma \) and F is the residue field of R modulo its ideal \(\gamma R\), i.e. \(F = R / \gamma R\), \(\vert F \vert = q\) with \(q=p^{\alpha }\) for some prime number p and let \(R^{\times }\) denote the multiplicative group of units of R. For any \(\omega \in R^{\times }\) and \(t \ge \lceil \frac{s}{2}\rceil \), the structural properties and dual codes of \((1+ \omega \gamma ^{t})\)-quasi-twisted (QT) codes of length \(n=\ell m\), with \((m, p)=1\), over R are given. The key idea is to view a \((1 + \omega \gamma ^{t})\)-QT code over R as a linear code over \(R_{m} = R[x] / \langle x^{m} - (1 + \omega \gamma ^{t}) \rangle \). Furthermore, given the decomposition of a \((1+ \omega \gamma ^{t})\)-QT code, we provide the decomposition of its dual code. As a result, a characterization of self-dual \((1+ \omega \gamma ^{t})\)-QT codes over a finite chain ring R, with \((1+ \omega \gamma ^{t})=(1+ \omega \gamma ^{t})^{-1}\), is provided. By using the Chinese remainder theorem or the discrete Fourier transform, the ring \(R[x] / \langle x^{m} - (1 + \omega \gamma ^{t}) \rangle \) can be decomposed into a direct sum of finite chain rings. The inverse transform of the discrete Fourier transform produces a method for deriving \((1+ \omega \gamma ^{t})\)-QT codes from codes of lower lengths.

For a binary linear code ${\cal C}$ and a vector u, we define and study the overlap polynomial $F^{(u)}_{{\cal C}}(Z,X,D)$ which is a sum over the words in ${\cal C}$ of terms that takes into account the weight of the element of ${\cal C}$ and its overlap with the vector $u$. We obtain a generalized MacWilliams relation between $F^{(u)}_{{\cal C}}(Z,X,D)$ and $F^{(u)}_{{\cal C}^{\perp}}(Z,X,D)$ and a recursion equation for determining the overlap polynomials of a self-dual code in terms of overlap polynomials corresponding to vectors of lower weight. We apply the recursion to determine the weight enumerators of the cosets of the (putative) [72,36,16] self-dual code of Type II, up to a small number of parameters. The overlap polynomials are expressed in terms of covering polynomials $g^{(u)}_{{\cal C}}(Z,X)$ that are sums over the elements of ${\cal C}$ that cover $u$. The design strength $\delta({\cal C})$ is closely related to the covering polynomials and we use them to get a sharpened version of the Assmus--Mattson theorem. In many cases we are able to conclude that certain codes hold $\delta({\cal C})$-designs but no weight class holds a $1+\delta({\cal C})$-design. We study the design strength of self-dual, doubly even extremal codes and, in the case of length divisible by 24, improve the well-known estimate $\delta({\cal C})\geq 5$ to state that either $\delta({\cal C})\geq 7$ or $\delta({\cal C})=5$ and no weight class holds a 6-design; similar results hold in the case of other lengths. Examples are given using a self-dual code [32,16,8], certain cyclic subcodes of the punctured Reed--Muller code ${\cal R}(2,m)^*$, the first order RM-code, and a [16,9,4] code in which all six of the weight classes hold 2-designs but one nontrivial weight class also holds a 3-design.

<abstract><p>In this paper, we consider $ S $-chains of extremal self-dual and self-orthogonal codes and their applications in the construction of quantum codes. Then, by virtue of covering radius, we determine necessary conditions for linear codes to have subcodes with large dual distances and design a new $ S $-chain search method. As computational results, 18 $ S $-chains with large distances are obtained, and many good quantum codes can be derived from those $ S $-chains by Steane construction, some of which improve the previous results.</p></abstract>

In this paper, we give optimalself-dual codes over $GF(5)$ for lengths $24$, $40$, $48$ and $56$. In particular,new inequivalent $[48, 24]$ and $[56, 28]$ self-dual codes over $GF(5)$ whose minimum weights are $14$ and $16$, are constructed usingskew-Hadamard matrices of order $24$ and $28$,thus improving the only known quadratic double circulant self-dual codes of length $48$ and $56$.Moreover, $[80, 40]$ and $[88, 44]$ self-dual codeswhose minimum weights are $17$ and $19$ over $GF(5)$, areconstructed for the first time. These codes are derived fromskew-Hadamard matrices of order $40$ and $44$, respectively. Finally, a new $[56, 28, 17]$ self-dual code is constructedover $GF(7)$ having the highest minimum weight among $[56, 28]$ self-dual codes.This new optimal code is constructed from a skew-Hadamard-matrix of order $28$, for the first time.

Self-dual codes over the Galois ring GR(4,2) are investigated. Of special interest are quadratic double circulant codes. Euclidean self-dual (Type II) codes yield self-dual (Type II) ℤ4-codes by projection on a trace orthogonal basis. Hermitian self-dual codes also give self-dual ℤ4-codes by the cubic construction, as well as Eisenstein lattices by Construction A. Applying a suitable Gray map to self-dual codes over the ring gives formally self-dual 𝔽4-codes, most notably in length 12 and 24. Extremal unimodular lattices in dimension 38, 42 and the first extremal 3-modular lattice in dimension 44 are constructed.

We obtain a characterization on self-orthogonality for a given binary linear code in terms of the number of column vectors in its generator matrix, which extends the result of Bouyukliev et al. (2006). As an application, we give an algorithmic method to embed a given binary k-dimensional linear code <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> ( k = 3,4) into a self-orthogonal code of the shortest length which has the same dimension k and minimum distance d' ≥ d( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> ). For k > 4, we suggest a recursive method to embed a k-dimensional linear code to a self-orthogonal code. We also give new explicit formulas for the minimum distances of optimal self-orthogonal codes for any length n with dimension 4 and any length n \not ≡ 6, 13,14,21,22,28,29 (mod 31) with dimension 5.