Mixed Alphabets Research Articles

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

Skew Cyclic and Skew Constacyclic Codes over a Mixed Alphabet

In this note, we study skew cyclic and skew constacyclic codes over the mixed alphabet R=FqR1R2, where q=pm, p is an odd prime with m odd and R1=Fq+uFq with u2=u, and R2=Fq+uFq+vFq with u2=u,v2=v,uv=vu=0. Such codes consist of the juxtaposition of three codes of the same size over Fq,R1, and R2, respectively. We investigate the generator polynomial for skew cyclic codes over R. Furthermore, we discuss the structural properties of the skew cyclic and skew constacyclic codes over R. We also study their q-ary images under suitable Gray maps.

Near MDS and near quantum MDS codes via orthogonal arrays

Near maximum distance separable (NMDS) codes are closely related to interesting objects in finite geometry and have nice applications in combinatorics and cryptography. But there are many unsolved problems about construction of NMDS codes. In this paper, by using symmetrical orthogonal arrays (OAs), we construct a lot of NMDS, m-MDS and almost extremal NMDS codes. Quantum error-correcting codes (QECCs) play a central role in quantum information processing and can protect quantum information from various quantum noises. We present a general method for constructing QECCs over mixed alphabets through asymmetrical OAs. Since quantum maximum distance separable (QMDS) codes over mixed alphabets with the dimension equal to one have not been found in all the literature so far, the definition of a near QMDS code over mixed alphabets is proposed. By using asymmetrical OAs, we obtain many such codes.

Z2[u]Z2[u]-Additive codes

In this paper, we study mixed alphabet codes over two alphabets [Formula: see text], and its extension [Formula: see text], [Formula: see text]. We define new Gray maps, using which we study cyclic codes and obtain the properties of their generators. Further, we construct constacyclic codes, and linear codes, and observe the useful properties of their generators. Finally, we present some examples of cyclic codes and binary linear codes.

Two classes of few-Lee weight [formula omitted]-linear codes using simplicial complexes and minimal codes via Gray map

Recently certain few-Lee weight additive codes over some mixed alphabet rings with optimal or minimal Gray images are constructed using suitable defining sets. Inspired by these works, we construct two special classes of linear codes over Z2[u] with u2=0 by using the mixed alphabet ring Z2Z2[u] and employing simplicial complexes having one or more maximal elements. We show that these codes have few-Lee weights by determining their Lee weight distributions. As applications, we are getting the 1st order binary Reed-Muller codes of length 23m−2 (where m>1) and some infinite families of binary self-orthogonal minimal codes via Gray map.

On Optimal and Quantum Code Construction from Cyclic Codes over FqPQ with Applications.

The key objective of this paper is to study the cyclic codes over mixed alphabets on the structure of FqPQ, where P=Fq[v]⟨v3-α22v⟩ and Q=Fq[u,v]⟨u2-α12,v3-α22v⟩ are nonchain finite rings and αi is in Fq/{0} for i∈{1,2}, where q=pm with m≥1 is a positive integer and p is an odd prime. Moreover, with the applications, we obtain better and new quantum error-correcting (QEC) codes. For another application over the ring P, we obtain several optimal codes with the help of the Gray image of cyclic codes.

Quantum Error-Correcting Codes Based on Orthogonal Arrays.

In this paper, by using the Hamming distance, we establish a relation between quantum error-correcting codes ((N,K,d+1))s and orthogonal arrays with orthogonal partitions. Therefore, this is a generalization of the relation between quantum error-correcting codes ((N,1,d+1))s and irredundant orthogonal arrays. This relation is used for the construction of pure quantum error-correcting codes. As applications of this method, numerous infinite families of optimal quantum codes can be constructed explicitly such as ((3,s,2))s for all si≥3, ((4,s2,2))s for all si≥5, ((5,s,3))s for all si≥4, ((6,s2,3))s for all si≥5, ((7,s3,3))s for all si≥7, ((8,s2,4))s for all si≥9, ((9,s3,4))s for all si≥11, ((9,s,5))s for all si≥9, ((10,s2,5))s for all si≥11, ((11,s,6))s for all si≥11, and ((12,s2,6))s for all si≥13, where s=s1⋯sn and s1,…,sn are all prime powers. The advantages of our approach over existing methods lie in the facts that these results are not just existence results, but constructive results, the codes constructed are pure, and each basis state of these codes has far less terms. Moreover, the above method developed can be extended to construction of quantum error-correcting codes over mixed alphabets.

Galois LCD codes over mixed alphabets

In this work we give a characterization of Galois Linear Complementary Dual codes and Galois-invariant codes over mixed alphabets of finite chain rings, which leads to the study of the Gray image of FpFp[θ]-linear codes, where p∈{2;3} and θ≠θ2=0 that provides LCD codes over Fp.

On a class of skew constacyclic codes over mixed alphabets and applications in constructing optimal and quantum codes

On a class of skew constacyclic codes over mixed alphabets and applications in constructing optimal and quantum codes

Quantum codes from cyclic codes over the mixed alphabet structure

Quantum codes from cyclic codes over the mixed alphabet structure

On skew cyclic codes over a mixed alphabet and their applications to DNA codes

In this paper, we introduce skew cyclic codes over the mixed alphabet [Formula: see text], where [Formula: see text] is the finite field with 4 elements and [Formula: see text]. Our results include a description of the generator polynomials of such codes and a necessary and sufficient condition for an [Formula: see text]-skew cyclic code to be reversible complement.

Constacyclic codes over mixed alphabets and their applications in constructing new quantum codes

Let p be an odd prime and m be a positive integer, $$q=p^m$$ , $${\mathcal {R}}={\mathbb {F}}_q+u{\mathbb {F}}_q$$ with $$u^2=u$$ , and $${\mathcal {S}}={\mathbb {F}}_q+u{\mathbb {F}}_q+v{\mathbb {F}}_q$$ with $$u^2=u, v^2=v, uv=vu=0$$ and $$\Lambda =(\lambda _1,\lambda _2,\lambda _3)\in {\mathbb {F}}_q{\mathcal {R}}{\mathcal {S}}$$ . In this paper, we study the algebraic structure of constacyclic codes over $${\mathcal {R}}$$ and $${\mathcal {S}}$$ . Further, we discuss the structure of $${\mathbb {F}}_q{\mathcal {R}}{\mathcal {S}}$$ - $$\Lambda $$ -constacyclic codes of block length $$(\alpha ,\beta ,\gamma )$$ . This family of codes can be viewed as $${\mathcal {S}}[x]$$ -submodules of $$\frac{{\mathbb {F}}_q[x]}{\langle x^{\alpha }-\lambda _1\rangle }\times \frac{{\mathcal {R}}[x]}{\langle x^{\beta }-\lambda _2\rangle }\times \frac{{\mathcal {S}}[x]}{\langle x^{\gamma }-\lambda _3\rangle }$$ . The generator polynomials of this family of codes are discussed. As application, we discuss the construction of quantum error-correcting codes (QECCs) from constacyclic codes over $${\mathbb {F}}_q{\mathcal {R}}{\mathcal {S}}$$ and obtain several new QECCs from this study.

MANUSCRIPT EVIDENCE FOR ALPHABET-SWITCHING IN THE WORKS OF CICERO: PROPER NOUNS AND ADJECTIVES

Our manuscripts of Cicero contain dozens of Greek words that are presented in some passages in Greek letters, and in others are transliterated into Latin. In a recent paper I collected the evidence for this phenomenon in connection with common nouns and adjectives (for example ὑποθήκη vs hypotheca, ἱστορικός vs historicus), surveyed scholarship to date and posited an interpretative framework which is assumed in this study also. Key components of this framework are the use of mixed alphabets in surviving ancient documents (especially inscriptions) and an awareness of the frequency with which modern editors change the alphabets in the manuscripts when dealing with Greek words—hence the importance of using the apparatus critici, not just the printed text, of our editions. The earlier paper was also strict in its exclusion of words in continuous passages, and even short phrases, of Greek, since that context excludes the option of transliteration for the author. The major contention of that earlier study was that a coherent pattern of use in the manuscripts can only really be a reflection of Cicero's own considered choice of alphabets: consistently inexplicable choice may indicate that Cicero himself was indifferent to which alphabet he used for single Greek words, or that our copyists paid no attention to this aspect of their exemplars, or both.

New DNA Codes from Cyclic Codes over Mixed Alphabets

Let R=F4+uF4,withu2=u and S=F4+uF4+vF4,withu2=u,v2=v,uv=vu=0. In this paper, we study F4RS-cyclic codes of block length (α,β,γ) and construct cyclic DNA codes from them. F4RS-cyclic codes can be viewed as S[x]-submodules of Fq[x]⟨xα−1⟩×R[x]⟨xβ−1⟩×S[x]⟨xγ−1⟩. We discuss their generator polynomials as well as the structure of separable codes. Using the structure of separable codes, we study cyclic DNA codes. By using Gray maps ψ1 from R to F42 and ψ2 from S to F43, we give a one-to-one correspondence between DNA codons of the alphabets {A,T,G,C}2,{A,T,G,C}3 and the elements of R,S, respectively. Then we discuss necessary and sufficient conditions of cyclic codes over F4, R, S and F4RS to be reversible and reverse-complement. As applications, we provide examples of new cyclic DNA codes constructed by our results.

Linear codes over 𝔽4R and their MacWilliams identity

Let [Formula: see text] be the field of four elements. We denote by [Formula: see text] the commutative ring, with [Formula: see text] elements, [Formula: see text] with [Formula: see text]. This work defines linear codes over the ring of mixed alphabets [Formula: see text] as well as their dual codes under a nondegenerate inner product. We then derive the systematic form of the respective generator matrices of the codes and their dual codes. We wrap the paper up by proving the MacWilliams identity for linear codes over [Formula: see text].

Z 4 Z 4 [u]-ADDITIVE CYCLIC AND CONSTACYCLIC CODES

We study mixed alphabet cyclic and constacyclic codes over the two alphabets Z 4 , the ring of integers modulo 4, and its quadratic extension Z 4 [u] = Z 4 + uZ 4 , u 2 = 0. Their generator polynomials and minimal spanning sets are obtained. Further, under new Gray maps, we nd cyclic, quasi-cyclic codes over Z 4 as the Gray images of both λ-constacyclic and skew λ-constacyclic codes over Z 4 [u]. Moreover, it is proved that the Gray images of Z 4 Z 4 [u]-additive constacyclic and skew Z 4 Z 4 [u]-additive constacyclic codes are generalized quasi-cyclic codes over Z 4. Finally, several new quaternary linear codes are obtained from these cyclic and constacyclic codes.

On $Z_p Z_{p^k}$ -Additive Codes and Their Duality

In this paper, two different Gray-like maps from $Z_p^\alpha\times Z_{p^k}^\beta$, where $p$ is prime, to $Z_p^n$, $n={\alpha+\beta p^{k-1}}$, denoted by $\phi$ and $\Phi$, respectively, are presented. We have determined the connection between the weight enumerators among the image codes under these two mappings. We show that if $C$ is a $Z_p Z_{p^k}$-additive code, and $C^\bot$ is its dual, then the weight enumerators of the image $p$-ary codes $\phi(C)$ and $\Phi(C^\bot)$ are formally dual. This is a partial generalization of [On $Z_{2^k}$-dual binary codes, arXiv:math/0509325], and the result is generalized to odd characteristic $p$ and mixed alphabet. Additionally, a construction of $1$-perfect additive codes in the mixed $Z_p Z_{p^2} ... Z_{p^k}$ alphabet is given.

The structure of ℤ2ℤ2s-additive cyclic codes

[Formula: see text]-additive codes for any integer [Formula: see text] are considered as codes over mixed alphabets. They are a generalization of binary linear codes and linear codes over [Formula: see text] In this paper, we are interested in studying [Formula: see text]-additive cyclic codes. We will give the generator polynomials of these codes. We will also give the minimal spanning sets for these codes. We will define separable [Formula: see text]-additive codes and provide conditions on the generator polynomials for a [Formula: see text]-additive cyclic code to be separable. Finally, we present some examples of optimal parameter binary codes obtained as images of [Formula: see text]-additive cyclic codes.

Variable strength covering arrays

AbstractWe define and study variable strength covering arrays (also called covering arrays on hypergraphs), which are generalizations of covering arrays and covering arrays on graphs. Variable strength covering arrays have the potential for use in software testing, allowing the engineer to omit the parameter combinations known to not interact in order to reduce the number of tests required. The present paper shows that variable strength covering arrays are relevant combinatorial objects that have deep connections with hypergraph homomorphisms and generalize other important combinatorial designs. We give optimal constructions for special types of hypergraphs, constructions based on columns with uniform occurrence of symbols, and constructions for mixed alphabets.

On the structure of [formula omitted]-linear and cyclic codes

Recently some special type of mixed alphabet codes that generalize the standard codes has attracted much attention. Besides Z2Z4-additive codes, Z2Z2[u]-linear codes are introduced as a new member of such families. In this paper, we are interested in a new family of such mixed alphabet codes, i.e., codes over Z2Z2[u3] where Z2[u3]={0,1,u,1+u,u2,1+u2,u+u2,1+u+u2} is an 8-element ring with u3=0. We study and determine the algebraic structures of linear and cyclic codes defined over this family. First, we introduce Z2Z2[u3]-linear codes and give standard forms of generator and parity-check matrices and later we present generators of both cyclic codes and their duals over Z2Z2[u3]. Further, we present some examples of optimal binary codes which are obtained through Gray images of Z2Z2[u3]-cyclic codes.