On distance Gray codes

A Gray code of size n is a cyclic sequence of all binary words of length n such that two consecutive words differ exactly in one position. We say that the Gray code is a distance code if the Hamming distance between words located at distance k from each other is equal to d. The distance property generalizes the familiar concepts of a locally balanced Gray code. We prove that there are no distance Gray codes with d = 1 for k > 1. Some examples of constructing distance Gray codes are given. For one infinite series of parameters, it is proved that there are no distance Gray codes.     

Symmetric LDPC codes and local testing

Coding theoretic and complexity theoretic considerations naturally lead to the question of generating symmetric, sparse, redundant linear systems. This paper provides a new way of construction with better parameters and new lower bounds.Low Density Parity Check (LDPC) codes are linear codes defined by short constraints (a property essential for local testing of a code). Some of the best (theoretically and practically) used codes are LDPC. Symmetric codes are those in which all coordinates “look the same,” namely there is some transitive group acting on the coordinates which preserves the code. Some of the most commonly used locally testable codes (especially in PCPs and other proof systems), including all “low-degree” codes, are symmetric. Requiring that a symmetric binary code of length n has large (linear or near-linear) distance seems to suggest a “con ict” between 1/rate and density (constraint length). In known constructions, if one is constant, then the other is almost the worst possible - n/poly(logn).Our main positive result simultaneously achieves symmetric low density, constant rate codes generated by a single constraint. We present an explicit construction of a symmetric and transitive binary code of length n, near-linear distance n/(log logn)², of constant rate and with constraints of length (logn)⁴. The construction is in the spirit of Tanner codes, namely the codewords are indexed by the edges of a sparse regular expander graph. The main novelty is in our construction of a transitive (non Abelian!) group acting on these edges which preserves the code. Our construction is one instantiation of a framework we call Cayley Codes developed here, that may be viewed as extending zig-zag product to symmetric codes.Our main negative result is that the parameters obtained above cannot be significantly improved, as long as the acting group is solvable (like the one we use). More specifically, we show that in constant rate and linear distance codes (aka “good” codes) invariant under solvable groups, the density (length of generating constraints) cannot go down to a constant, and is bounded below by (log^(Ω(?)) n)(an Ω(?) iterated logarithm) if the group has a derived series of length ?. This negative result precludes natural local tests with constantly many queries for such solvable “good” codes.     

On separability of the functional space with the open-point and bi-point-open topologies

We study the property of separability of functional space C(X) with the open-point and bi-point-open topologies and show that it is consistent with ZFC that there is a set of reals of cardinality \({\mathfrak{c}}\) such that a set C(X) with the open-point topology is not a separable space. We also show in a set model (the iterated perfect set model) that for every set of reals X, C(X) with the bi-point-open topology is a separable space.     

On full-rank perfect codes over finite fields

We propose a construction of full-rank q-ary 1-perfect codes. This is a generalization of the construction of full-rank binary 1-perfect codes by Etzion and Vardy (1994). The properties of the i-components of q-ary Hamming codes are investigated, and the construction of full-rank q-ary 1-perfect codes is based on these properties. The switching construction of 1-perfect codes is generalized to the q-ary case. We propose a generalization of the notion of an i-component of a 1-perfect code and introduce the concept of an (i, σ)-component of a q-ary 1-perfect code. We also present a generalization of the Lindström–Schönheim construction of q-ary 1-perfect codes and provide a lower bound for the number of pairwise distinct q-ary 1-perfect codes of length n.     

Almost cover-free codes and designs

An s-subset of codewords of a binary code X is said to be \((s,\,\ell )\) -bad in X if the code X contains a subset of \(\ell \) other codewords such that the conjunction of the \(\ell \) codewords is covered by the disjunctive sum of the s codewords. Otherwise, the s-subset of codewords of X is called \((s,\,\ell )\) -good in X. A binary code X is said to be a cover-free (CF) \((s,\,\ell )\)-code if the code X does not contain \((s,\,\ell )\)-bad subsets. In this paper, we introduce a natural probabilistic generalization of CF \((s,\,\ell )\)-codes, namely: a binary code X is said to be an almost CF \((s,\,\ell )\)-code if the relative number of its \((s,\,\ell )\)-good s-subsets is close to 1. We develop a random coding method based on the ensemble of binary constant weight codes to obtain lower bounds on the capacity of such codes. Our main result shows that the capacity for almost CF \((s,\,\ell )\)-codes is essentially greater than the rate for ordinary CF \((s,\,\ell )\)-codes.     

Euclidean and Hermitian LCD MDS codes

Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual is trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. On the other hand, being optimal codes, maximum distance separable codes (abbreviated MDS) are of much interest from many viewpoints due to their theoretical and practical properties. However, little work has been done on LCD MDS codes. In particular, determining the existence of q-ary [n, k] LCD MDS codes for various lengths n and dimensions k is a basic and interesting problem. In this paper, we firstly study the problem of the existence of q-ary [n, k] LCD MDS codes and solve it for the Euclidean case. More specifically, we show that for \(q>3\) there exists a q-ary [n, k] Euclidean LCD MDS code, where \(0\le k \le n\le q+1\), or, \(q=2^{m}\), \(n=q+2\) and \(k= 3 \text { or } q-1\). Secondly, we investigate several constructions of new Euclidean and Hermitian LCD MDS codes. Our main techniques in constructing Euclidean and Hermitian LCD MDS codes use some linear codes with small dimension or codimension, self-orthogonal codes and generalized Reed-Solomon codes.     

Structural properties and enumeration of 1-generator generalized quasi-cyclic codes

Let F _q be a finite field of cardinality q, m ₁, m ₂, . . . , m _l be any positive integers, and \({A_i=F_q[x]/(x^{m_i}-1)}\) for i = 1, . . . , l. A generalized quasi-cyclic (GQC) code of block length type (m ₁, m ₂, . . . , m _l ) over F _q is defined as an F _q [x]-submodule of the F _q [x]-module \({A_1\times A_2\times\cdots\times A_l}\). By the Chinese Remainder Theorem for F _q [x] and enumeration results of submodules of modules over finite commutative chain rings, we investigate structural properties of GQC codes and enumeration of all 1-generator GQC codes and 1-generator GQC codes with a fixed parity-check polynomial respectively. Furthermore, we give an algorithm to count numbers of 1-generator GQC codes.     

On the minimum size of binary codes with length 2<Emphasis Type="Italic">R</Emphasis> + 4 and covering radius <Emphasis Type="Italic">R</Emphasis>

The minimum size of a binary code with length n and covering radius R is denoted by K(n, R). For arbitrary R, the value of K(n, R) is known when n ≤  2R +  3, and the corresponding optimal codes have been classified up to equivalence. By combining combinatorial and computational methods, several results for the first open case, K(2R +  4, R), are here obtained, including a proof that K(10, 3) =  12 with 11481 inequivalent optimal codes and a proof that if K(2R +  4, R) <  12 for some R then this inequality cannot be established by the existence of a corresponding self-complementary code.     

Infinite families of 3-designs from a type of five-weight code

It has been known for a long time that t-designs can be employed to construct both linear and nonlinear codes and that the codewords of a fixed weight in a code may hold a t-design. While a lot of progress in the direction of constructing codes from t-designs has been made, only a small amount of work on the construction of t-designs from codes has been done. The objective of this paper is to construct infinite families of 2-designs and 3-designs from a type of binary linear codes with five weights. The total number of 2-designs and 3-designs obtained in this paper are exponential in any odd m and the block size of the designs varies in a huge range.     

On complementary dual quasi-twisted codes

A linear complementary-dual (LCD) code C is a linear code whose dual code \(C^{\perp }\) satisfies \(C \cap C^{\perp }=\{0\}\). In this work we characterize some classes of LCD q-ary \((\lambda , l)\)-quasi-twisted (QT) codes of length \(n=ml\) with \((m,q)=1\), \(\lambda \in F_{q} \setminus \{0\}\) and \(\lambda \ne \lambda ^{-1}\). We show that every \((\lambda ,l)\)-QT code C of length \(n=ml\) with \(dim(C)<m\) or \(dim(C^{\perp })<m\) is an LCD code. A sufficient condition for r-generator QT codes is provided under which they are LCD. We show that every maximal 1-generator \((\lambda ,l)\)-QT code of length \(n=ml\) with \(l>2\) is either an LCD code or a self-orthogonal code and a sufficient condition for this family of codes is given under which such a code C is LCD. Also it is shown that every maximal 1-generator \((\lambda ,2)\)-QT code is LCD. Several good and optimal LCD QT codes are presented.     

The Uncountable Lattice of All Subfunctors of the Functor of Rational Representations

In set theory the cardinality of the continuum \(|{\mathbb R}|\) is the cardinal number of some interesting sets, like the Cantor set or the transcendental numbers. We will prove that the cardinal number of all subfunctors of the functor of rational representations \(k \otimes_{{\mathbb Z}} R_{{\mathbb Q}}\), taking values on odd order groups over the field k of characteristic 2, is equal to \(|{\mathbb R}|\). When the characteristic q?>?0 of the field k is not necessarily even, we will present a formula giving the dimension of the evaluations S _C,k(G), of the simple functor S _C,k, at any group G of order prime to q and being associated to a suitable cyclic group C.     

An upper bound for the number of uniformly packed binary codes

Under study are the binary codes uniformly packed (in the wide sense) of length n with minimum distance d and covering radius ρ. It is shown that every code of this kind is uniquely determined by the set of its codewords of weights ?n/2? ? ρ, …, ?n/2? + ρ. For odd d, the number of distinct codes is at most

$2^{2^{n - \tfrac{3}{2}\log n + o(log n)} } $

.

     

Approximations and Mittag-Leffler conditions the applications

A classic result by Bass says that the class of all projective modules is covering if and only if it is closed under direct limits. Enochs extended the if-part by showing that every class of modules C, which is precovering and closed under direct limits, is covering, and asked whether the converse is true. We employ the tools developed in [18] and give a positive answer when C = A, or C is the class of all locally A^≤ω-free modules, where A is any class of modules fitting in a cotorsion pair (A, B) such that B is closed under direct limits. This setting includes all cotorsion pairs and classes of locally free modules arising in (infinite-dimensional) tilting theory. We also consider two particular applications: to pure-semisimple rings, and Artin algebras of infinite representation type.     

On Functions with Zero Integral Moments over Balls of Fixed Radius

We consider precise conditions ensuring that a function having zero integrals over all balls of fixed radius is equal to zero. The case where, together with zero integrals for f over congruent balls, the functions x_jf for all j have zero integrals over these balls is completely investigated. An inversion formula recovering a function in the class C^∞ from indicated integral moments is derived.     

On needed reals

Given a binary relationR, we call a subsetA of the range ofR R-adequate if for everyx in the domain there is someyεA such that (x, y)εR. Following Blass [4], we call a realη ”needed” forR if in everyR-adequate set we find an element from whichη is Turing computable. We show that every real needed for inclusion on the Lebesgue null sets,Cof(\(\mathcal{N}\)), is hyperarithmetic. Replacing “R-adequate” by “R-adequate with minimal cardinality” we get the related notion of being “weakly needed”. We show that it is consistent that the two notions do not coincide for the reaping relation. (They coincide in many models.) We show that not all hyperarithmetic reals are needed for the reaping relation. This answers some questions asked by Blass at the Oberwolfach conference in December 1999 and in [4].     

Dirichlet and Neumann problems for Laplace and heat equations in domains with right angles

The Dirichlet and Neumann problems are considered in the n-dimensional cube and in a right angle. The right-hand side is assumed to be bounded, and the boundary conditions are assumed to be zero. The author obtains a priori bounds for solutions in the Zygmund space, which is wider than the Lipschitz space C ^1,1 but narrower than the Hölder space C ^{1, α}, 0 < α < 1. Also, the first and second boundary-value problems are considered for the heat equation with similar conditions. It is shown that the solutions belong to the corresponding Zygmund space.     

On locally balanced gray codes

We consider locally balanced Gray codes.We say that a Gray code is locally balanced if every “short” subword in its transition sequence contains all letters of the alphabet |1, 2,..., n~. The minimal length of these subwords is the window width of the code. We show that for each n ≥ 3 there exists a Gray code with window width at most n + 3?log n?.     

Linear codes over $$\mathbb {F}_{q}[x]/(x^2)$$ and $$GR(p^2,m)$$GR(p2,m) reaching the Griesmer bound

We construct two series of linear codes C(G) over \(\mathbb {F}_{q}[x]/(x^2)\) and \(GR(p^2,m)\) reaching the Griesmer bound. Moreover, we consider the Gray images of C(G). The results show that the Gray images of C(G) over \(\mathbb {F}_{q}[x]/(x^2)\) are linear and also reach the Griesmer bound in some cases, and many of linear codes over \(\mathbb {F}_{q}\) we constructed have two Hamming (non-zero) weights.