On linear subspace codes closed under intersection

IEEE Transactions on Information Theory, 2000

The projective space of order n over the finite field q , denoted here as Pq(n), is the set of all subspaces of the vector space n q . The projective space can be endowed with the distance function d(U; V ) = dim U + dim V 0 2 dim(U \V ) which turns Pq(n) into a metric space. With this, an (n; M; d) code in projective space is a subset of Pq(n) of size M such that the distance between any two codewords (subspaces) is at least d. Koetter and Kschischang recently showed that codes in projective space are precisely what is needed for error-correction in networks: an (n; M; d) code can correct t packet errors and packet erasures introduced (adversarially) anywhere in the network as long as 2t + 2 < d. This motivates our interest in such codes. In this paper, we investigate certain basic aspects of "coding theory in projective space." First, we present several new bounds on the size of codes in P q (n), which may be thought of as counterparts of the classical bounds in coding theory due to Johnson, Delsarte, and Gilbert-Varshamov. Some of these are stronger than all the previously known bounds, at least for certain code parameters. We also present several specific constructions of codes and code families in P q (n). Finally, we prove that nontrivial perfect codes in P q (n) do not exist.

Given an (n, k) linear code C over GF(q), the intersection of C with a code ?(C), where ? # S n , is an (n, k 1 ) code, where max[0, 2k&n] k 1 k. The intersection problem is to determine which integers in this range are attainable for a given code C. We show that, depending on the structure of the generator matrix of the code, some of the values in this range are attainable. As a consequence we give a complete solution to the intersection problem for most of the interesting linear codes, e.g. cyclic codes, Reed Muller codes, and most MDS codes.

Finite Fields and Their Applications, 2018

Consider the Grassmann graph formed by k-dimensional subspaces of an n-dimensional vector space over the field of q elements (1 < k < n − 1) and denote by Π(n, k)q the restriction of this graph to the set of projective [n, k]q codes. In the case when q ≥ n 2 , we show that the graph Π(n, k)q is connected, its diameter is equal to the diameter of the Grassmann graph and the distance between any two vertices coincides with the distance between these vertices in the Grassmann graph. Also, we give some observations concerning the graphs of simplex codes. For example, binary simplex codes of dimension 3 are precisely maximal singular subspaces of a non-degenerate quadratic form.

Discrete Mathematics, 1985

We study pairs of binary linear codes Cl(n, nR1), C2(n, nR 2) with the property that for any nonzero cl c C~ and c2~ C 2, there are coordinates in which both c, and c 2 are nonzero. * This work was done while he was visiting at the Technion, Haifa, Israel. 0012-365X/85/$3.30 (~) 1985, Elsevier Science Publishers B.V. (North-Holland)

Contemporary Mathematics, 2010

The finite projective space PG(n, q), q = p h , p prime, h ≥ 1, is also investigated from a coding-theoretical point of view. The linear code Cs,t(n, q) of s-spaces and tspaces in a projective space PG(n, q), q = p h , p prime, h ≥ 1, is defined as the vector space spanned over Fp by the rows of the incidence matrix of s-spaces and t-spaces. This linear code can be investigated purely for its coding-theoretical importance, but the properties of this linear code are also of interest for the finite projective space PG(n, q) itself. Some of the best results on substructures of finite projective spaces PG(n, q) have been obtained by using their corresponding codes. Recently, there has been a new incentive on the study of the minimum distance of these linear codes and their duals. In this paper, we summarize what is currently known about the minimum distance and small weight codewords of these linear codes and their duals.

IEEE Transactions on Information Theory, 2009

Coding in the projective space has received recently a lot of attention due to its application in network coding. Reduced row echelon form of the linear subspaces and Ferrers diagram can play a key role for solving coding problems in the projective space. In this paper we propose a method to design error-correcting codes in the projective space. We use a multilevel approach to design our codes. First, we select a constant weight code. Each codeword defines a skeleton of a basis for a subspace in reduced row echelon form. This skeleton contains a Ferrers diagram on which we design a rank-metric code. Each such rank-metric code is lifted to a constant dimension code. The union of these codes is our final constant dimension code. In particular the codes constructed recently by Koetter and Kschischang are a subset of our codes. The rank-metric codes used for this construction form a new class of rank-metric codes. We present a decoding algorithm to the constructed codes in the projective space. The efficiency of the decoding depends on the efficiency of the decoding for the constant weight codes and the rank-metric codes. Finally, we use puncturing on our final constant dimension codes to obtain large codes in the projective space which are not constant dimension.

arXiv (Cornell University), 2018

In this paper, a linear ℓ-intersection pair of codes is introduced as a generalization of linear complementary pairs of codes. Two linear codes are said to be a linear ℓ-intersection pair if their intersection has dimension ℓ. Characterizations and constructions of such pairs of codes are given in terms of the corresponding generator and parity-check matrices. Linear ℓ-intersection pairs of MDS codes over F q of length up to q + 1 are given for all possible parameters. As an application, linear ℓ-intersection pairs of codes are used to construct entanglement-assisted quantum error correcting codes. This provides a large number of new MDS entanglement-assisted quantum error correcting codes.

arXiv (Cornell University), 2021

Linear codes in the projective space P q (n), the set of all subspaces of the vector space F n q , were first considered by Braun, Etzion and Vardy. The Grassmannian G q (n, k) is the collection of all subspaces of dimension k in P q (n). We study equidistant linear codes in P q (n) in this paper and establish that the normalized minimum distance of a linear code is maximum if and only if it is equidistant. We prove that the upper bound on the size of such class of linear codes is 2 n when q = 2 as conjectured by Braun et al. Moreover, the codes attaining this bound are shown to have structures akin to combinatorial objects, viz. Fano plane and sunflower. We also prove the existence of equidistant linear codes in P q (n) for any prime power q using Steiner triple system. Thus we establish that the problem of finding equidistant linear codes of maximum size in P q (n) with constant distance 2d is equivalent to the problem of finding the largest d-intersecting family of subspaces in G q (n, 2d) for all 1 ≤ d ≤ n 2. Our discovery proves that there exist equidistant linear codes of size more than 2 n for every prime power q > 2.

arXiv (Cornell University), 2015

The critical exponent of a matroid is one of the important parameters in matroid theory and is related to the Rota and Crapo's Critical Problem. This paper introduces the covering dimension of a linear code over a finite field, which is analogous to the critical exponent of a representable matroid. An upper bound on the covering dimension is conjectured and nearly proven, improving a classical bound for the critical exponent. Finally, a construction is given of linear codes that attain equality in the covering dimension bound.

Advances in Mathematics, 2007

In this paper we prove that a set of points (in a projective space over a finite field of q elements), which is incident with 0 mod r points of every hyperplane, has at least (r −1)q +(p−1)r points, where 1 < r < q = p h , p prime. An immediate corollary of this theorem is that a linear code whose weights and length have a common divisor r < q and whose dual minimum distance is at least 3, has length at least (r − 1)q + (p − 1)r. The theorem, which is sharp in some cases, is a strong generalisation of an earlier result on the non-existence of maximal arcs in projective planes; the proof involves polynomials over finite fields, and is a streamlined and more transparent version of the earlier one.

Advances in Mathematics of Communications, 2011

Constant dimension codes, with a prescribed minimum distance, have found recently an application in network coding. All the codewords in such a code are subspaces of F n q with a given dimension. A computer search for large constant dimension codes is usually inefficient since the search space domain is extremely large. Even so, we found that some constant dimension lexicodes are larger than other known codes. We show how to make the computer search more efficient. In this context we present a formula for the computation of the distance between two subspaces, not necessarily of the same dimension.

In this paper we construct, using GAP System for Computational Discrete Algebra, some cyclic subspace codes, specially an optimal code over the finite field F 2 10. Further we introduce the q-analogous of a m-quasi cyclic subspace code over finite fields.

arXiv (Cornell University), 2024

The hull of a linear code C is the intersection of C with its dual code. We present and analyze the number of linear q-ary codes of the same length and dimension but with different dimensions for their hulls. We prove that for given dimension k and length n ≥ 2k the number of all [n, k] q linear codes with hull dimension l decreases as l increases. We also present classification results for binary and ternary linear codes with trivial hulls (LCD and self-orthogonal) for some values of the length n and dimension k, comparing the obtained numbers with the number of all linear codes for the given n and k.

Discrete Applied Mathematics, 2019

By an optimal linear code we mean that it has the highest minimum distance with a prescribed length and dimension. We construct several families of optimal linear codes over the finite field F p by making use of down-sets generated by one maximal element of F n p. Moreover, we show that these families of optimal linear codes are minimal and contain relative two-weight linear codes, and have applications to secret sharing schemes and wire-tap channel of type II with the coset coding scheme, respectively.

2011