The hull of a linear code over finite fields is the intersection of the code and its dual, and linear codes with small hulls have applications in computational complexity and information protection. Linear codes with the smallest hull are LCD codes, which have been widely studied. Recently, several papers were devoted to related LCD codes over finite fields with size greater than 3 to linear codes with one-dimensional or higher dimensional hull. Therefore, an interesting and non-trivial problem is to study binary linear codes with one-dimensional hull with connection to binary LCD codes. The objective of this paper is to study some properties of binary linear codes with one-dimensional hull, and establish their relation with binary LCD codes. Some interesting inequalities are thus obtained. Using such a characterization, we study the largest minimum distance $d_{one}(n,k)$ among all binary linear $[n,k]$ codes with one-dimensional hull. We determine the largest minimum distances $d_{one}(n,n-k)$ for $ k\leq 5$ and $d_{one}(n,k)$ for $k\leq 4$ or $14\leq n\leq 24$. We partially determine the exact value of $d_{one}(n,k)$ for $k=5$ or $25\leq n\leq 30$.
Linear codes with small hulls over finite fields have been extensively studied due to their practical applications in computational complexity and information protection. In this paper, we develop a general method to determine the exact value of $D_4^H(n,k,1)$ for $n\leq 12$ or $k\in \{1,2,3,n-1,n-2,n-3\}$, where $D_4^H(n,k,1)$ denotes the largest minimum distance among all quaternary linear $[n,k]$ codes with one-dimensional Hermitian hull. As a consequence, we solve a conjecture proposed by Mankean and Jitman on the largest minimum distance of a quaternary linear code with one-dimensional He…
We prove that any Hermitian self-orthogonal $[n,k,d]_{q^2}$ code gives rise to an $[n,k,d]_{q^2}$ code with $\ell$ dimensional Hermitian hull for $0\le \ell \le k$. We present a new method to construct Hermitian self-orthogonal $[n,k]_{q^2}$ codes with large dimensions $k>\frac{n+q-1}{q+1}$. New families of Hermitian self-orthogonal codes with good parameters are obtained; more precisely those containing almost MDS codes. By applying a puncturing technique to Hermitian self-orthogonal codes, MDS $[n,k]_{q^2}$ linear codes with Hermitian hull having large dimensions $k>\frac{n+q-1}{q+1}$ are al…
The hull of a linear code over finite fields is the intersection of the code and its dual, which was introduced by Assmus and Key. In this paper, we develop a method to construct linear codes with trivial hull ( LCD codes) and one-dimensional hull by employing the positive characteristic analogues of Gauss sums. These codes are quasi-abelian, and sometimes doubly circulant. Some sufficient conditions for a linear code to be an LCD code (resp. a linear code with one-dimensional hull) are presented. It is worth mentioning that we present a lower bound on the minimum distances of the constructed …
In this paper, we give a method for constructing linear codes with small hulls by generalizing the method in \cite{LCD-T-matric}. As a result, we obtain many optimal Euclidean LCD codes and Hermitian LCD codes, which improve the previously known lower bound on the largest minimum distance. We also obtain many optimal codes with one-dimension hull. Furthermore, we give three tables about formally self-dual LCD codes.
The Euclidean hull of a linear code $C$ is the intersection of $C$ with its Euclidean dual $C^\perp$. The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a linear code and checking permutation equivalence of two linear codes. The Euclidean hull of a linear code has been applied to the so-called entanglement-assisted quantum error-correcting codes (EAQECCs) via classical error-correcting codes. In this paper, we consider linear codes with one-dimensional Euclidean hull from algebraic geometry…
In this work, we study and determine the dimensions of Euclidean and Hermitian hulls of two classical propagation rules, namely, the direct sum construction and the $(\mathbf{u},\mathbf{u+v})$-construction. Some new criteria for the resulting codes derived from these two propagation rules being self-dual, self-orthogonal, or linear complementary dual (LCD) codes are given. As an application, we construct some linear codes with prescribed hull dimensions, many new binary, ternary Euclidean formally self-dual (FSD) LCD codes, and quaternary Hermitian FSD LCD codes. Some new even-like, odd-like, …
The intersection ${\bf C}\bigcap {\bf C}^{\perp}$ (${\bf C}\bigcap {\bf C}^{\perp_h}$) of a linear code ${\bf C}$ and its Euclidean dual ${\bf C}^{\perp}$ (Hermitian dual ${\bf C}^{\perp_h}$) is called the Euclidean (Hermitian) hull of this code. It is natural to consider the hull-variation problem when a linear code ${\bf C}$ is transformed to an equivalent code ${\bf v} \cdot {\bf C}$. In this paper we introduce the maximal hull dimension as an invariant of a linear code with respect to the equivalent transformations. Then some basic properties of the maximal hull dimension are studied. We p…
The hull $H(C)$ of a linear code $C$ is defined by $H(C)=C \cap C^\perp$. A linear code with a complementary dual (LCD) is a linear code with $H(C)=\{0\}$. The dimension of the hull of a code is an invariant under permutation equivalence. For binary and ternary codes the dimension of the hull is also invariant under monomial equivalence and we show that this invariant is determined by the extended weight enumerator of the code.\\ The hull of a code is not invariant under monomial equivalence if $q\geq 4$. We show that every ${\mathbb F}_q $-linear code is monomial equivalent with an LCD code i…
This paper proposes new propagation rules on quantum codes in the entanglement-assisted and in quantum subsystem scenarios. The rules lead to new families of such quantum codes whose parameters are demonstrably optimal. To obtain the results, we devise tools to puncture and shorten codes in ways that ensure their Hermitian hulls have certain desirable properties. More specifically, we give a general framework to construct $k$-dimensional generalized Reed-Solomon codes whose Hermitian hulls are $(k-1)$-dimensional maximum distance separable codes.
We introduce the notion of quadratic hull of a linear code, and give some of its properties. We then show that any symmetric bilinear multiplication algorithm for a finite-dimensional algebra over a field can be obtained by evaluation-interpolation at simple points (i.e. of degree and multiplicity 1) on a naturally associated space, namely the quadratic hull of the corresponding code. This also provides a geometric answer to some questions such as: which linear maps actually are multiplication algorithms, or which codes come from supercodes (as asked by Shparlinski-Tsfasman-Vladut). We illustr…
We analyze the orthogonal greedy algorithm when applied to dictionaries $\mathbb{D}$ whose convex hull has small entropy. We show that if the metric entropy of the convex hull of $\mathbb{D}$ decays at a rate of $O(n^{-\frac{1}{2}-α})$ for $α> 0$, then the orthogonal greedy algorithm converges at the same rate on the variation space of $\mathbb{D}$. This improves upon the well-known $O(n^{-\frac{1}{2}})$ convergence rate of the orthogonal greedy algorithm in many cases, most notably for dictionaries corresponding to shallow neural networks. These results hold under no additional assumptions on…
This paper presents a construction for several families of optimal binary locally repairable codes (LRCs) with small locality (2 and 3). This construction is based on various anticodes. It provides binary LRCs which attain the Cadambe-Mazumdar bound. Moreover, most of these codes are optimal with respect to the Griesmer bound.
A general construction of binary sequences with low autocorrelation are considered in the paper. Based on recent progresses about this topic and this construction, several classes of binary sequences with optimal autocorrelation and other low autocorrelation are presented.
Binary sequences with optimal autocorrelation play important roles in radar, communication, and cryptography. Finding new binary sequences with optimal autocorrelation has been an interesting research topic in sequence design. Ding-Helleseth-Lam sequences are such a class of binary sequences of period $p$, where $p$ is an odd prime with $p\equiv 1(\bmod~4)$. The objective of this letter is to present a construction of binary sequences of period $4p$ via interleaving four suitable Ding-Helleseth-Lam sequences. This construction generates new binary sequences with optimal autocorrelation which c…
The binary Hamming codes with parameters $[2^m-1, 2^m-1-m, 3]$ are perfect. Their extended codes have parameters $[2^m, 2^m-1-m, 4]$ and are distance-optimal. The first objective of this paper is to construct a class of binary linear codes with parameters $[2^{m+s}+2^s-2^m,2^{m+s}+2^s-2^m-2m-2,4]$, which have better information rates than the class of extended binary Hamming codes, and are also distance-optimal. The second objective is to construct a class of distance-optimal binary codes with parameters $[2^m+2, 2^m-2m, 6]$. Both classes of binary linear codes have new parameters.
Binary sequences with optimal autocorrelation and large linear complexity have important applications in cryptography and communications. Very recently, a class of binary sequences of period $4p$ with optimal autocorrelation was proposed via interleaving four suitable Ding-Helleseth-Lam sequences (Des. Codes Cryptogr., DOI 10.1007/s10623-017-0398-5), where $p$ is an odd prime with $p\equiv 1(\bmod~4)$. The objective of this paper is to determine the minimal polynomial and the linear complexity of this class of binary optimal sequences via sequence polynomial approach. It turns out that this cl…
We propose a simple modification to the recently proposed compressive binary search. The modification removes an unnecessary and suboptimal factor of log log n from the SNR requirement, making the procedure optimal (up to a small constant). Simulations show that the new procedure performs significantly better in practice as well. We also contrast this problem with the more well known problem of noisy binary search.
A new [48,16,16] optimal linear binary block code is given. To get this code a general construction is used which is also described in this paper. The construction of this new code settles an conjecture mentioned in a 2008 paper by Janosov et al. where the authors found an new optimal [47,15,16]-code, which is relevant to the applied construction.
Recently a construction of optimal non-constant dimension subspace codes, also termed projective space codes, has been reported in a paper of Honold-Kiermaier-Kurz. Restricted to binary codes in a 5-dimensional ambient space with minimum subspace distance 3, these optimal codes were interpreted in terms of maximal partial spreads of 2-dimensional subspaces. In a parallel development, an optimal binary (5,3) code was obtained by a minimal change strategy on a nearly optimal example of Etzion and Vardy. In this article, we report several examples of optimal binary (5,3) codes obtained by the app…
In steganography, we always hope to maximize the embedding payload subject to an upper-bounded distortion. We need suitable distortion measurement to evaluate the embedding impact. However, different distortion functions exposes different levels of distortion evaluation, implying that we have different optimization distributions by applying different distortion functions. In applications, the embedding distortion is caused by a certain number of embedding operations. Instead of a predefined distortion, we actually utilize a number of modifications to embed as many message bits as possible as l…
Compressed sensing is a promising technique that attempts to faithfully recover sparse signal with as few linear and nonadaptive measurements as possible. Its performance is largely determined by the characteristic of sensing matrix. Recently several zero-one binary sensing matrices have been deterministically constructed for their relative low complexity and competitive performance. Considering the complexity of implementation, it is of great practical interest if one could further improve the sparsity of binary matrix without performance loss. Based on the study of restricted isometry proper…
In this letter, the optimality of the likelihood ratio test (LRT) is investigated for binary hypothesis testing problems in the presence of a behavioral decision-maker. By utilizing prospect theory, a behavioral decision-maker is modeled to cognitively distort probabilities and costs based on some weight and value functions, respectively. It is proved that the LRT may or may not be an optimal decision rule for prospect theory based binary hypothesis testing and conditions are derived to specify different scenarios. In addition, it is shown that when the LRT is an optimal decision rule, it corr…
Recently, Chang and Hyun obtained some classes of binary optimal codes via simplicial complexes. In this letter, we utilize posets of the disjoint union of two chains to construct binary optimal linear codes.
Constructions of binary sequences with low autocorrelation are considered in the paper. Based on recent progresses about this topic, several more general constructions of binary sequences with optimal autocorrelations and other low autocorrelations are presented.
We consider upward scaling an overloaded min-TSC binary signature set and propose an optimal solution based on improved sphere decoding algorithm. Instead of previous suboptimal approach, we are guaranteed to find the optimal maximum-likelihood (ML) searching result with low complexity.
A noisy entropy inequality for boolean functions by Samorodnitsky is applied to binary codes. It is shown that a binary code that achieves capacity on the binary erasure channel admits optimal list size for list decoding on some binary symmetric channels (in a regime where this optimal list size is exponentially large).
