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The rank of adjacency matrix plays an important role in construction of linear codes from a directed strongly regular graph using different techniques, namely, code orthogonality, adjacency matrix determinant and adjacency matrix spectrum. The problem of computing the dimensions of such codes is an intriguing one. Several conjectures to determine the rank of adjacency matrix of a DSRG Γ over a finite field, keep researchers working in this area. To address the same to an extent, we have considered the problem of finding the rank over a finite field of the adjacency matrix of a DSRG Γ (v , k , t , λ , μ) with μ = 1 , including some mixed Moore graphs and corresponding codes arising from them, in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel class of real number codes termed as ℝ-spectral codes, derived from real symmetric operators. Our focus is on addressing the challenges associated with error correction in communication systems utilizing real number codes. Our study initiates with a meticulous definition and analysis of the fundamental properties of ℝ-spectral codes. These properties serve as the foundation for comprehending the distinctive characteristics that ℝ-spectral codes offer, ensuring the integrity and reliability of transmitted data. Our exploration extends to the construction of maximum distance separable (MDS) ℝ-spectral codes, which boast the maximum error correction capability. As a culmination of our research, we present an efficient decoding technique tailored for ℝ-spectral codes. Leveraging an MDS ℝ-spectral code and our decoding technique, we achieve optimal error correction and ensure good numerical stability. This innovative approach opens new avenues for enhancing the performance of real-number codes in practical applications. [ABSTRACT FROM AUTHOR]

The error-correcting pair is a general algebraic decoding method for linear codes. Since every linear code is contained in an MDS linear code with the same minimum distance over some finite field extensions, we focus on MDS linear codes. Recently, He and Liao showed that for an MDS linear code 풞 with minimum distance 2ℓ + 2, if it has an ℓ-error-correcting pair, then the parameters of the pair have three possibilities. Moreover, for the first case, they gave a necessary condition for an MDS linear code 풞 with minimum distance 2ℓ + 2 to have an ℓ-error-correcting pair, and for the other two cases, they only gave some counterexamples. For the second case, in this paper, we give a necessary condition for an MDS linear code 풞 with minimum distance 2ℓ + 2 to have an ℓ-error-correcting pair, and then basing on the Product Singleton Bound, we prove that there are two cases for such pairs, and then give some counterexamples basing on twisted generalized Reed–Solomon codes for these cases. [ABSTRACT FROM AUTHOR]

Secret-sharing is a vital topic of cryptography and has widely used in information protection. One technique to the development of secret-sharing schemes is primarily based on error-correcting codes. In this paper, we determine the minimal codewords of relative four-weight codes and have the utility in the secret sharing schemes. [ABSTRACT FROM AUTHOR]

In this paper, a channel-optimized quantum communication scheme with imperfect Bell states is being considered, based on quantum teleportation and error correction codes technology. In this scheme, a novel "combination code" technique is proposed to protect the pre-shared Bell states from storage errors and to safeguard the information-encoded states from noisy channels. Ultimately, it is revealed by Monte Carlo simulation results that the channel fidelity of the proposed channel-optimized quantum communication scheme with imperfect Bell states is significantly advantageous over standard quantum error-correcting code (QECC) and entanglement-assisted quantum error-correcting code (EAQECC) communication schemes. [ABSTRACT FROM AUTHOR]

In this paper, we use a method inspired by the valuation theory to classify completely cyclic codes of length 3 p s over p m + u p m + u 2 p m . Then, we compute their separation parameters which was useful to compute the Hamming distance in [8]. [ABSTRACT FROM AUTHOR]

Let R be the ring F q [ u , v , w ] / 〈 u 2 − 1 , v 2 − 1 , w 3 − w , u v − v u , v w − w v , w u − u w 〉 , where q = p m for any odd prime p and positive integer m. In this paper, we study constacyclic codes over the ring R. We define a Gray map by a matrix and decompose a constacyclic code over the ring R as the direct sum of constacyclic codes over F q , we also characterize self-dual constacyclic codes over the ring R and give necessary and sufficient conditions for constacyclic codes to be dual-containing. As an application, we give a method to construct quantum codes from dual-containing constacyclic codes over the ring R. [ABSTRACT FROM AUTHOR]

Any integral convex polytope P in ℝ N provides an N -dimensional toric variety X P and an ample divisor D P on this variety. This paper gives an explicit construction of the algebraic geometric error-correcting code on X P , obtained by evaluating global section of the line bundle corresponding to D P on every rational point of X P . This work presents an extension of toric codes analogous to the one of Reed–Muller codes into projective ones, by evaluating on the whole variety instead of considering only points with nonzero coordinates. The dimension of the code is given in terms of the number of integral points in the polytope P and an algorithmic technique to get a lower bound on the minimum distance is described. [ABSTRACT FROM AUTHOR]

In this paper, we propose quantum stabilizer codes based on a new construction of self-orthogonal trace-inner product codes over the Galois field with 4 elements (GF(4)). First, from any two binary vectors, we construct a generator matrix of linear codes whose components are over GF(4). We prove that the proposed linear codes comply with the self-orthogonal, trace-inner product. Then, we propose mapping tables to construct new quantum stabilizer codes by using linear codes. Comparison results show that our proposed quantum codes have various dimensions for any code length with the capacity for better errors correction relative to the referenced quantum codes. [ABSTRACT FROM AUTHOR]

A novel approach for construction of quantum codes employing jacket transform is proposed in this paper. The constructed quantum nested codes has less complexities than the previous quantum codes in encoding procedures as well as decoding procedures. Furthermore, the quantum error codes can be constructed with an arbitrary size using the number decomposition, and the corresponding distance of constructed quantum codes can be adjusted by changing decomposition methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of computing all possible order ideals and also sets connected to 1, and the corresponding border bases, for the vanishing ideal of a given finite set of points. In this context, two different approaches are discussed: based on the Buchberger–Möller Algorithm [H. M. Möller and B. Buchberger, The construction of multivariate polynomials with preassigned zeros, EUROCAM '82 Conf., Computer Algebra, Marseille/France 1982, Lect. Notes Comput. Sci. 144, (1982), pp. 24–31], we first propose a new algorithm to compute all possible order ideals and the corresponding border bases for an ideal of points. The second approach involves adapting the Farr–Gao Algorithm [J. B. Farr and S. Gao, Computing Gröbner bases for vanishing ideals of finite sets of points, in 16th Int. Symp. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC-16, Las Vegas, NV, USA (Springer, Berlin, 2006), pp. 118–127] for finding all sets connected to 1, as well as the corresponding border bases, for an ideal of points. It should be noted that our algorithms are term ordering free. Therefore, they can compute successfully all border bases for an ideal of points. Both proposed algorithms have been implemented and their efficiency is discussed via a set of benchmarks. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new complex Fibonacci matrix H p , n whose elements are complex Fibonacci numbers and we developed a new coding and decoding method followed from this complex Fibonacci matrix H p , n . We establish the relations among the code matrix elements, error detection and correction for this coding theory. [ABSTRACT FROM AUTHOR]

Etzion [T. Etzion, Perfect byte correcting codes, IEEE Trans. Inform. Theory44 (1998) 3140–3146.] has classified byte error correcting binary codes into five different categories with respect to the size of the byte. If a code is partitioned into m equal byte of size say β then n = m β or the size of bytes is n m . Alternatively if bytes are of different size say, β 1 , β 2 , ... , β m then n = ∑ i = 1 m β i . The result was further modified by Tyagi and Sethi [V. Tyagi and A. Sethi, b i -Byte correcting perfect codes, Asian-Eur. J. Math.7(1) (2014) 1–8.] and the classification of bytes was given with respect to size of the byte as well as length of the burst. We call such codes as b i -byte correcting perfect codes. Our aim in this paper is to find the possibilities for the existence of b i -byte correcting non-binary perfect codes. [ABSTRACT FROM AUTHOR]