Optimal Binary Codes Research Articles

Infinite Families of Few Weight Optimal Binary Linear Codes From Multivariable Functions

Infinite Families of Few Weight Optimal Binary Linear Codes From Multivariable Functions

Optimal Binary Few-Weight Codes Using a Mixed Alphabet Ring and Simplicial Complexes

Optimal Binary Few-Weight Codes Using a Mixed Alphabet Ring and Simplicial Complexes

New constructions of optimal binary LCD codes

Linear complementary dual (LCD) codes provide an optimum linear coding solution for the two-user binary adder channel. LCD codes also can be used against side-channel attacks and fault non-invasive attacks. In this paper, we obtain a lower bound on the distance of binary LCD codes through expanded codes. We give necessary and sufficient conditions to extend binary [n,k] LCD codes to binary [n+1,k] and [n+1,k+1] LCD codes. A sufficient condition for a binary [n,k,d−1] LCD code constructed from a binary [n+1,k,d]LCDo,e code is also given. Finally, we construct some new binary LCD codes by using our LCD bounds and some methods such as puncturing, shortening, expanding and extension. In particular, we improve some previously known range of dLCD(n,k) of lengths 38≤n≤40 and dimensions 9≤k≤15. We also obtain some values or range of dLCD(n,k) with 41≤n≤50 and 6≤k≤n−6.

Minimal and optimal binary codes obtained using $$C_D$$-construction over the non-unital ring I

Minimal and optimal binary codes obtained using $$C_D$$-construction over the non-unital ring I

A Polynomial Time Algorithm for Constructing Optimal Binary AIFV-2 Codes

Huffman Codes are <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">optimal</i> Instantaneous Fixed-to-Variable (FV) codes in which every source symbol can only be encoded by one codeword. Relaxing these constraints permits constructing better FV codes. More specifically, recent work has shown that AIFV- <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> codes can beat Huffman coding. AIFV- <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> codes construct an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> -tuple of different coding trees between which the code alternates and are only <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">almost instantaneous</i> (AI). This means that decoding a word might require a delay of a finite number of bits. Current algorithms for constructing optimal AIFV- <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> codes are iterative processes that construct progressively “better sets” of code trees. The processes have been proven to finitely converge to the optimal code but with no known bounds on the convergence rate. This paper derives a geometric interpretation of the space of binary AIFV-2 codes, permitting the development of the first polynomially time-bounded procedure for constructing optimal AIFV codes. This binary-search like procedure will run in <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</i> ) time, where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> is the number of symbols in the source alphabet and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</i> is the maximum number of bits used to encode any one input probability.

On Toeplitz codes of index t and isometry codes

In this paper, we introduce Toeplitz codes of index t, which are a generalization of quasi-cyclic codes of index t. We show that such codes meet the asymptotic Gilbert-Varshamov bound on the relative distance with asymptotic rate 1t over Fq, so they are asymptotically good for any t. We characterize some special classes of Toeplitz codes as LCD codes (resp. linear codes with one-dimensional hull or self-orthogonal codes). It is worth mentioning that we determine the hull dimension of the isometric image of a linear code. Some optimal or almost optimal binary linear codes are obtained as a result. Most importantly, we obtain some optimal or almost optimal LCD codes (resp. linear codes with one-dimensional hull) from quasi-cyclic codes. In particular, we construct binary LCD [32,16,8], [33,11,11] and [34,17,8] codes with improved parameters directly.

Construction of quasi-cyclic self-dual codes over finite fields

ABSTRACT Our goal of this paper is to find a construction of all ℓ-quasi-cyclic self-dual codes over a finite field F q of length mℓ for every positive even integer ℓ. In this paper, we study the case where x m − 1 has an arbitrary number of irreducible factors in F q [ x ] ; in the previous studies, only some special cases where x m − 1 has exactly two or three irreducible factors in F q [ x ] , were studied. Firstly, the binary code case is completed: for any even positive integer ℓ, every binary ℓ-quasi-cyclic self-dual code can be obtained by our construction. Secondly, we work on the q-ary code cases for an odd prime power q. We find an explicit method for construction of all ℓ-quasi-cyclic self-dual codes over F q of length mℓ for any even positive integer ℓ, where we require that q ≡ 1 ( mod 4 ) if the index ℓ ≥ 6 . By implementation of our method, we obtain a new optimal binary self-dual code [ 172 , 86 , 24 ] , which is also a quasi-cyclic code of index 4.

Duadic codes over Fp + uFp + vFp + uvFp

Duadic codes constitute a well-known class of cyclic codes. In this paper, we study the structure of duadic codes of length n over the ring R = Fp + uFp + vFp + uvFp, u2 = v2 = 0, uv = vu, where p is prime and (n, p) = 1. These codes have been studied here in the setting of abelian codes over R, and we have used Fourier transform and idempotents to study them. We have characterized abelian codes over R by studying their torsion and residue codes. It is shown that the Gray image of an abelian code of length n over R is a binary abelian code of length 4n. Conditions for self-duality and self-orthogonality of duadic codes over R are derived. Some conditions on the existence of self-dual augmented and extended codes over R are presented. We have also studied Type II self-dual augmented and extended codes over R. Some results related to theminimum Lee distances of duadic codes over R are presented. We have also presented a sufficient condition for abelian codes of the same length over R to have the same minimum Hamming distance. Some optimal binary linear codes of length 36 and ternary linear codes of length 16 have been obtained as Gray images of duadic codes of length 9 and 4, respectively, over R using the computational algebra system Magma.

Enumeration of Optimal Equidistant Codes

Problems of search and enumeration of binary and ternary equidistant codes are considered in the paper. We investigate some combinatorial algorithms and develop specialized computer packages to find non-equivalent optimal binary and ternary equidistant codes for 3 ≤ d ≤ n ≤ 9.

Optimal binary linear codes from posets of the disjoint union of two chains

&lt;p style='text-indent:20px;'&gt;Recently, some infinite families of optimal binary linear codes are constructed from simplicial complexes. Afterwards, the construction method was extended to using arbitrary posets. In this paper, based on a generic construction of linear codes, we obtain four classes of optimal binary linear codes by using the posets of two chains. Two of them induce Griesmer codes which are not equivalent to the linear codes constructed by Belov. Those codes are exploited to construct secret sharing schemes in cryptography as well.&lt;/p&gt;

Selection on Golay complementary sequences in binary pulse compression for microbubble detection

In medical ultrasound imaging using microbubbles (MBs), the nonlinear echoes from the MBs are used for contrast-specific image construction. Techniques such as pulse inversion, amplitude modulation, and the combined method (PIAM) are employed to increase nonlinear components in the echoes. In addition, employment of pulse compression using binary-coded ultrasound can potentially increase the components. In the case of the nonlinear echo, however, a nonlinear sidelobe occurs around the compressed pulse (correlation peak), and nonlinear components for PIAM in the correlation peak are reduced. The shape of the nonlinear sidelobe and the nonlinear-component reduction in the correlation peak can be estimated from the binary codes used. In this study, the optimal binary codes for PIAM are determined from all patterns of 10, 16, and 20 bit Golay codes. Then, the performance of PIAM with pulse compression using each code is evaluated via computer simulations and experiments using the SonoVue MBs.

On Constructions of One-Lee Weight Codes Over Z₄

Let $\mathbb {Z}_{4}$ be the integer ring of residue classes modulo 4. In this paper, we construct four infinite families of $\mathbb {Z}_{4}$ -codes with one nonzero Lee weight by their generator matrices. Furthermore, we study the linearity of their Gray images and obtain a family of optimal binary codes.

Binary Code Optimized for Partial Encryption

A common technique for the partial encryption of compressed images and videos encrypts only the sign bits of some syntax elements such as the quantized transform coefficients or the motion vector differences. The sign bit can be interpreted as the most significant bit (MSB) in the binary representation of the syntax element. Our work is motivated by the key observation that the binary code used for this representation has an impact on the quality of the reconstruction at the eavesdropper and on the size of the stream to be encrypted. Therefore, we address the problem of optimal binary code design for partial encryption. Ideally, the goal is to simultaneously maximize the eavesdropper’s distortion and minimize the length of the compressed MSB stream. Since these two objectives are conflicting in general, we formulate the problem as the maximization of a weighted sum of the eavesdropper’s distortion and of the probability of the MSB being 0. We cast the problem as a binary integer linear program equivalent to a maximum weight matching problem, which has a polynomial-time solution algorithm. We show that when the source to be quantized and the quantizer are symmetric, the problem can be converted to a linear program of a smaller size, for a family of distortion metrics. Extensive experiments assess the performance of the optimized binary code in comparison with existing approaches. The results reveal that certain existing partial encryption schemes could benefit from the proposed design.

A New Family of Optimal Binary Few-Weight Codes From Simplicial Complexes

Trace codes over a cubic ring extension of the binary field are constructed by using a triple of simplicial complexes. A linear Gray map yields several infinite families of binary few-weight codes which can have as many as 9 weights. By equating two complexes in the triple, we obtain binary three-weight codes, which are minimal and distance optimal.

Optimal minimal linear codes from posets

Recently, some infinite families of minimal and optimal binary linear codes were constructed from simplicial complexes by Hyun et al. We extend this construction method to arbitrary posets. Especially, anti-chains are corresponded to simplicial complexes. In this paper, we present two constructions of binary linear codes from hierarchical posets of two levels. In particular, we determine the weight distributions of binary linear codes associated with hierarchical posets with two levels. Based on these results, we also obtain some optimal and minimal binary linear codes not satisfying the condition of Ashikhmin–Barg.

Self-Orthogonal Codes Constructed from Posets and Their Applications in Quantum Communication

It is an important issue to search for self-orthogonal codes for construction of quantum codes by CSS construction (Calderbank-Sho-Steane codes); in quantum error correction, CSS codes are a special type of stabilizer codes constructed from classical codes with some special properties, and the CSS construction of quantum codes is a well-known construction. First, we employ hierarchical posets with two levels for construction of binary linear codes. Second, we find some necessary and sufficient conditions for these linear codes constructed using posets to be self-orthogonal, and we use these self-orthogonal codes for obtaining binary quantum codes. Finally, we obtain four infinite families of binary quantum codes for which the minimum distances are three or four by CSS construction, which include binary quantum Hamming codes with length n≥7. We also find some (almost) “optimal” quantum codes according to the current database of Grassl. Furthermore, we explicitly determine the weight distributions of these linear codes constructed using posets, and we present two infinite families of some optimal binary linear codes with respect to the Griesmer bound and a class of binary Hamming codes.

Infinite Families of Optimal Linear Codes Constructed From Simplicial Complexes

A linear code is optimal if it has the highest minimum distance of any linear code with a given length and dimension. We construct infinite families of optimal binary linear codes C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δc</sub> constructed from simplicial complexes in F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , where Δ is a simplicial complex in F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> and Δ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sup> the complement of Δ. We first find an explicit computable criterion for C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δc</sub> to be optimal; this criterion is given in terms of the 2-adic valuation of Σ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j=1</sub> 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|Ai|-1</sup> , where the At's are maximal elements of Δ. Furthermore, we obtain much simpler criteria under various specific conditions on the maximal elements of Δ. In particular, we find that C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δc</sub> is a Griesmer code if and only if the maximal elements of Δ are pairwise disjoint and their sizes are all distinct. Specially, when f has exactly two maximal elements, we explicitly determine the weight distribution of C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δc</sub> .We present many optimal linear codes constructed by our method, and we emphasize that we obtain at least 32 new optimal linear codes.

Optimal binary and ternary linear codes with hull dimension one

Hulls of linear codes have been of interest and extensively studied due to their wide applications. In this paper, we focus on constructions and optimality of linear codes with hull dimension one over small finite fields. General constructions for such codes are given together with the analysis on their parameters. Optimal linear $$[n,2,d]_q$$ codes with hull dimension one are presented for all positive integers $$n\ge 3$$ and $$q\in \{2,3\}$$ . Moreover, for $$q=2$$ , the enumeration of such optimal codes is given up to equivalence.

Optimal binary codes derived from $$\mathbb {F}_{2} \mathbb {F}_4$$-additive cyclic codes

In this paper, we study the algebraic structure of additive cyclic codes over the alphabet $${\mathbb {F}}_{2}^{r}\times {\mathbb {F}}_{4}^{s}={ \mathbb {F}}_{2}^{r}{\mathbb {F}}_{4}^{s},$$ where r and s are non-negative integers, $$\mathbb {F}_{2}={\mathbb {GF}}(2)$$ and $$\mathbb {F}_{4}={\mathbb {GF}} (4)$$ are the finite fields of 2 and 4 elements, respectively. We determine generator polynomials for $$\mathbb {F}_{2}\mathbb {F}_{4}$$ -additive cyclic codes. We also introduce a linear map W that depends on the trace map T to relate these codes to binary linear codes over $$\mathbb {F} _{2}.$$ Further, we investigate the duals of $$\mathbb {F}_{2}\mathbb {F}_{4}$$ -additive cyclic codes. We show that the dual of any $$\mathbb {F}_{2}\mathbb {F }_{4}$$ -additive cyclic code is another $$\mathbb {F}_{2}\mathbb {F}_{4}$$ -additive cyclic code. Using the mapping W, we provide examples of $$\mathbb {F}_{2}\mathbb {F}_{4}$$ -additive cyclic codes whose binary images have optimal parameters. We also consider additive cyclic codes over $$\mathbb {F}_{4}$$ and give some examples of optimal parameter quantum codes over $$\mathbb {F}_{4}$$ .

Cyclic codes over $${\mathbb {F}}_2 +u{\mathbb {F}}_2+v{\mathbb {F}}_2 +v^2 {\mathbb {F}}_2 $$ with respect to the homogeneous weight and their applications to DNA codes

In this paper, we study cyclic codes and their duals over the local Frobenius non-chain ring $$R={\mathbb {F}}_2[u,v] / \langle u^2=v^2,uv \rangle $$, and we obtain optimal binary linear codes with respect to the homogeneous weight over R via a Gray map. Moreover, we characterize DNA codes as images of cyclic codes over R.