A subspace code is a nonempty set of subspaces of a vector space $\mathbb F^n_q$. Linear codes with complementary duals, or LCD codes, are linear codes whose intersection with their duals is trivial. In this paper, we introduce a notion of LCD subspace codes. We show that the minimum distance decoding problem for an LCD subspace code reduces to a problem that is simpler than for a general subspace code. Further, we show that under some conditions equitable partitions of association schemes yield such LCD subspace codes and as an illustration of the method give some examples from distance-regular graphs. We also give a construction from mutually unbiased Hadamard matrices, and more generally, from mutually unbiased weighing matrices.
Linear codes with complementary duals are linear codes whose intersection with their duals are trivial, shortly named LCD codes. In this paper we outline a construction for LCD codes over finite fields of order $q$ using weighing matrices and their orbit matrices. The LCD codes constructed can be of any length dimension according to the choice of matrices used in their construction. As a special case, LCD codes of length $2n$ and dimension $n$ are constructed which also have the property of being formally self-dual. Alternatively, under a condition depending on $q$ that the codes are not LCD, …
We give two methods for constructing many linear complementary dual (LCD for short) codes from a given LCD code, by modifying some known methods for constructing self-dual codes. Using the methods, we construct binary LCD codes and quaternary Hermitian LCD codes, which improve the previously known lower bound on the largest minimum weights.
In the 2017 paper by Dougherty, Kim, Ozkaya, Sok, and Solé about the linear programming bound for LCD codes the notion $\mathrm{LCD}[n,k]$ was defined for binary LCD $[n,k]$-codes. We find the formula for $\mathrm{LCD}[n,2]$.
Linear complementary dual (LCD) codes are linear codes that intersect with their dual codes trivially. We study the largest minimum weight $d_2(n,k)$ among all binary LCD $[n,k]$ codes and the largest minimum weight $d_3(n,k)$ among all ternary LCD $[n,k]$ codes. The largest minimum weights $d_2(n,5)$ and $d_3(n,4)$ are partially determined. We also determine the largest minimum weights $d_2(n,n-5)$, $d_3(n,n-i)$ for $i \in \{2,3,4\}$, and $d_3(n,k)$ for $n \in \{11,12,\ldots,19\}$.
We propose a method for a classification of quaternary Hermitian LCD codes having large minimum weights. As an example, we give a classification of quaternary optimal Hermitian LCD codes of dimension 3.
Linear complementary dual codes (or codes with complementary duals) are codes whose intersections with their dual codes are trivial. These codes were first introduced by Massey in 1964. Nowadays, LCD codes are extensively studied in the literature and widely applied in data storage, cryptography, etc. In this paper, we prove some properties of binary LCD codes using their shortened and punctured codes. We also present some inequalities for the largest minimum weight $d_{LCD}(n,k)$ of binary LCD [n,k] codes for given length n and dimension k. Furthermore, we give two tables with the values of $…
We introduce a general construction of many Hermitian LCD $[n, k]$ codes from a given Hermitian LCD $[n, k]$ code. Furthermore, we present some results on punctured codes and shortened codes of quaternary Hermitian LCD codes. As an application, we improve some of the previously known lower bounds on the largest minimum weights of quaternary Hermitian LCD codes of length $12 \le n \le 30$.
Linear complementary dual (LCD) codes are linear codes that intersect with their dual trivially. We give a characterization of LCD codes over $\mathbb{F}_q$ having large minimum weights for $q \in \{2,3\}$. Using the characterization, we determine the largest minimum weights among LCD $[n,k]$ codes over $\mathbb{F}_q$ for $(q,k) \in \{(2,4), (3,2),(3,3)\}$. Moreover, we give a complete classification of optimal LCD $[n,k]$ codes over $\mathbb{F}_q$ for $(q,k) \in \{(2,3), (2,4), (3,2),(3,3)\}$.
From a given $[n, k]$ code $C$, we give a method for constructing many $[n, k]$ codes $C'$ such that the hull dimensions of $C$ and $C'$ are identical. This method can be applied to constructions of both self-dual codes and linear complementary dual codes (LCD codes for short). Using the method, we construct 661 new inequivalent extremal doubly even $[56, 28, 12]$ codes. Furthermore, constructing LCD codes by the method, we improve some of the previously known lower bounds on the largest minimum weights of binary LCD codes of length $n=26,28 \le n \le 40$.
Hermitian linear complementary dual codes are linear codes whose intersection with their Hermitian dual code is trivial. The largest minimum weight among quaternary Hermitian linear complementary dual codes of dimension $2$ is known for each length. We give the complete classification of optimal quaternary Hermitian linear complementary dual codes of dimension $2$.
For $k \ge 2$ and a positive integer $d_0$, we show that if there exists no quaternary Hermitian linear complementary dual $[n,k,d]$ code with $d \ge d_0$ and Hermitian dual distance greater than or equal to $2$, then there exists no quaternary Hermitian linear complementary dual $[n,k,d]$ code with $d \ge d_0$ and Hermitian dual distance $1$. As a consequence, we generalize a result by Araya, Harada and Saito on the nonexistence of some quaternary Hermitian linear complementary dual codes.
Subspace codes, i.e., sets of subspaces of $\mathbb{F}_q^v$, are applied in random linear network coding. Here we give improved upper bounds for their cardinalities based on the Johnson bound for constant dimension codes.
In this article, constant dimension subspace codes whose codewords have subspace distance in a prescribed set of integers, are considered. The easiest example of such an object is a {\it junta}; i.e. a subspace code in which all codewords go through a common subspace. We focus on the case when only two intersection values for the codewords, are assigned. In such a case we determine an upper bound for the dimension of the vector space spanned by the elements of a non-junta code. In addition, if the two intersection values are consecutive, we prove that such a bound is tight, and classify the ex…
We consider (symmetric, non-degenerate) bilinear spaces over a finite field and investigate the properties of their $\ell$-complementary subspaces, i.e., the subspaces that intersect their dual in dimension $\ell$. This concept generalizes that of a totally isotropic subspace and, in the context of coding theory, specializes to the notions of self-orthogonal, self-dual and linear-complementary-dual (LCD) codes. In this paper, we focus on the enumerative and asymptotic combinatorics of all these objects, giving formulas for their numbers and describing their typical behavior (rather than the be…
We investigate subspace codes whose codewords are subspaces of ${\rm PG}(4,q)$ having non-constant dimension. In particular, examples of optimal mixed-dimension subspace codes are provided, showing that ${\cal A}_q(5,3) = 2(q^3+1)$.
In the context of constant--dimension subspace codes, an important problem is to determine the largest possible size $A_q(n, d; k)$ of codes whose codewords are $k$-subspaces of $\mathbb{F}_q^n$ with minimum subspace distance $d$. Here in order to obtain improved constructions, we investigate several approaches to combine subspace codes. This allow us to present improvements on the lower bounds for constant--dimension subspace codes for many parameters, including $A_q(10, 4; 5)$, $A_q(12, 4; 4)$, $A_q(12, 6, 6)$ and $A_q(16, 4; 4)$.
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 present a definition and an example of the $q$-analogous of a $m$-quasi-cyclic subspace code over F_{2^{8}}.
Codes in finite projective spaces equipped with the subspace distance have been proposed for error control in random linear network coding. Here we collect the present knowledge on lower and upper bounds for binary subspace codes for projective dimensions of at most $7$. We obtain several improvements of the bounds and perform two classifications of optimal subspace codes, which are unknown so far in the literature.
Cyclic subspace codes gained a lot of attention especially because they may be used in random network coding for correction of errors and erasures. Roth, Raviv and Tamo in 2018 established a connection between cyclic subspace codes (with certain parameters) and Sidon spaces. These latter objects were introduced by Bachoc, Serra and Zémor in 2017 in relation with the linear analogue of Vosper's Theorem. This connection allowed Roth, Raviv and Tamo to construct large classes of cyclic subspace codes with one or more orbits. In this paper we will investigate cyclic subspace codes associated to a …
We improve on the lower bound of the maximum number of planes of ${\rm PG}(8,q)$ mutually intersecting in at most one point leading to the following lower bound: ${\cal A}_q(9, 4; 3) \ge q^{12}+2q^8+2q^7+q^6+q^5+q^4+1$ for constant dimension subspace codes. We also construct two new non-equivalent $(6, (q^3-1)(q^2+q+1), 4; 3)_q$ constant dimension subspace orbit-codes.
One of the main problems of the research area of network coding is to compute good lower and upper bounds of the achievable cardinality of so-called subspace codes in $\operatorname{PG}(n,q)$, i.e., the set of subspaces of $\mathbb{F}_q^n$, for a given minimal distance. Here we generalize a construction of Etzion and Silberstein to a wide range of parameters. This construction, named coset construction, improves or attains several of the previously best-known subspace code sizes and attains the MRD bound for an infinite family of parameters.
We revisit and extend the connections between $\mathbb{F}_{q^m}$-linear rank-metric codes and evasive $\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^m}^k$. We give a unifying framework in which we prove in an elementary way how the parameters of a rank-metric code are related to special geometric properties of the associated evasive subspace, with a particular focus on the generalized rank weights. In this way, we can also provide alternative and very short proofs of known results on scattered subspaces. We then use this simplified point of view in order to get a geometric characterization of near…
After a seminal paper by Shekeey (2016), a connection between maximum $h$-scattered $\mathbb{F}_q$-subspaces of $V(r,q^n)$ and maximum rank distance (MRD) codes has been established in the extremal cases $h=1$ and $h=r-1$. In this paper, we propose a connection for any $h\in\{1,\ldots,r-1\}$, extending and unifying all the previously known ones. As a consequence, we obtain examples of non-square MRD codes which are not equivalent to generalized Gabidulin or twisted Gabidulin codes. Up to equivalence, we classify MRD codes having the same parameters as the ones in our connection. Also, we deter…
