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Cyclic codes over R have been introduced recently. In this paper, we study the cyclic codes over R and their Z 2 image. Making use of algebraic structure, we find the some good Z 2 codes of length 28.
In this paper, we develop the theory for constructing DNA cyclic codes of odd length over R = Z 4 [u]/ u 2 -1 based on the deletion distance. Firstly, we relate DNA pairs with a special 16 elements of ring R. Cyclic codes of odd length... more
In this paper, we study the theory for constructing DNA cyclic codes of odd length over Z 4 [u]/ u 2 which play an important role in DNA computing. Cyclic codes of odd length over Z 4 + uZ 4 satisfy the reverse constraint and the... more
In this paper, we mainly study the some structure of cyclic DNA codes of odd length over the We established a direct link between the element of ring R and 64 codons by introducing a Gray map from R to R 1 = F 2 + uF 2 , u 2 = 1 where R 1... more
In this paper, we mainly study the some structure of cyclic DNA codes of odd length over the ring R = _2[u,v]/〈 u^2-1,v^3-v,uv-vu 〉 which play an important role in DNA computing. We established a direct link between the element of ring R... more
In this paper, we study the theory for constructing DNA cyclic codes of odd length over $\Z_4[u]/\langle u^2 \rangle$ which play an important role in DNA computing. Cyclic codes of odd length over $\Z_4 + u \Z_4$ satisfy the reverse... more
In this paper, we mainly study the some structure of cyclic DNA codes of odd length over the ring R = F2[u, v]/〈u 2 − 1, v3 − v, uv − vu〉 which play an important role in DNA computing. We established a direct link between the element of... more
Cyclic codes over R have been introduced recently. In this paper, we study the cyclic codes over R and their $\Z_2$ image. Making use of algebraic structure, we find the some good $\Z_2$ codes of length 28.
The use of skew polynomial rings allows to endow linear codes with cyclic structures which are not cyclic in the classical (commutative) sense. Whenever these skew cyclic structures are carefully chosen, some control over the Hamming... more
Warning. The Handbook of Coding Theory was published in 1998. Some research problems, presented as Open problems in this chapter, are solved, or partially solved, today.
The associated codes of almost perfect nonlinear (APN) functions have been widely studied. In this paper, we consider more generally the codes associated with functions that have differential uniformity at least 4. We emphasize, for such... more
Cyclically permutable codes have been studied for several applications involving synchronization, code-division multiple-access (CDMA) radio systems and optical CDMA. The usual emphasis is on finding constant weight cyclically permutable... more
1. Introduction. Let 9I(r) denote the automorphism group of the tournament T. Let g(n) be the maximum of |3I(r)| taken over all tournaments of order n. It was noted in [3] that g(n) is also the order of the subgroups of S n of maximum odd... more
Abs tract: OFDM have a several attractive features which give many advantages for transmission of fast and reliable data, high spectral efficiency, and easy equalization. During the transmission of this OFDM signal PAPR is a major... more
2" modified prime codes are designed for all-optical code-division multiple access (CDMA) networks using very simple encoders and decoders. The proposed code is obtained from an original 2" prime code of prime number P, by padding P ~ 1... more
The main objective of this research is to present the relationship between the subject of coding theory and three-dimensional projection space in the eighth field. We found the points, lines, and planes of the Galois field of order 8... more
In this work, a unique set of generators for a cyclic code over a finite chain ring has been established. The minimal spanning set and rank of the code have also been determined. Further, sufficient as well as necessary conditions for a... more
We study [2 m -1, 2m]-binary linear codes whose weights lie between w 0 and 2 m -w 0 , where w 0 takes the highest possible value. Primitive cyclic codes with two zeros whose dual satisfies this property actually correspond to almost bent... more
Let F 2 m be a finite field of cardinality 2 m , R = F 2 m + uF 2 m (u 2 = 0) and s, n be positive integers such that n is odd. In this paper, we give an explicit representation for every self-dual cyclic code over the finite chain ring R... more
Let F p m be a finite field of cardinality p m where p is an odd prime, n be a positive integer satisfying gcd(n, p) = 1, and denote R = F p m [u]/ u e where e ≥ 4 be an even integer. Let δ, α ∈ F × p m . Then the class of (δ + αu 2... more
I T ] 6 O ct 2 01 8 A class of repeated-root constacyclic codes over F p m [ u ] / 〈 u e 〉 of Type 2
Let Fpm be a finite field of cardinality p m where p is an odd prime, n be a positive integer satisfying gcd(n, p) = 1, and denote R = Fpm[u]/〈u〉 where e ≥ 4 be an even integer. Let δ, α ∈ Fpm. Then the class of (δ + αu)-constacyclic... more
Let F2m be a finite field of cardinality 2 m and s a positive integer. Using properties for Kronecker product of matrices and calculation for linear equations over F2m , an efficient method for the construction of all distinct self-dual... more
For any prime number p, positive integers m, k, n satisfying gcd(p, n) = 1 and λ 0 ∈ F × p m , we prove that any λ p k 0 -constacyclic code of length p k n over the finite field F p m is monomially equivalent to a matrix-product code of a... more
For any different odd primes ℓ and p, structure of constacyclic codes of length 2ℓ m p n over a finite field Fq of characteritic p and their duals is established in term of their generator polynomials. Among other results, all linear... more
For any different odd primes ℓ and p, structure of constacyclic codes of length 2ℓ m p n over a finite field Fq of characteritic p and their duals is established in term of their generator polynomials. Among other results, all linear... more
For any different odd primes l and p, structure of constacyclic codes of length 2lp over a finite field Fq of characteritic p and their duals is established in term of their generator polynomials. Among other results, all linear... more
In the above correspondence,' on page 1233, (6) is missing, on the right-hand side, a factor of This missing phase factor eventually cancels and the final published pdf is correct as it stands. (Alternatively, one can absorb the missing... more
We provide a construction for quantum codes (hermitian-self-orthogonal codes over GF (4)) starting from cyclic codes over GF (4 m ). We also provide examples of these codes some of which meet the known bounds for quantum codes.
We calculate the linear complexity of almost perfect binary sequences. Also we study the linear complexity of binary sequences obtained from series of almost perfect ternary sequences and the ternary sequences with two nonzero... more
We calculate the linear complexity of almost perfect binary sequences. Also we study the linear complexity of binary sequences obtained from series of almost perfect ternary sequences and the ternary sequences with two nonzero... more
For any prime p, all constacyclic codes of length p s over the ring R = F p m + uF p m are considered. The units of the ring R are of the forms γ and α + uβ, where α, β, and γ are nonzero elements of F p m , which provides p m (p m − 1)... more
The structures of cyclic and negacyclic codes of length and their duals over a finite chain ring are established when is not divisible by the characteristic of the residue field. Some cases where is divisible by the characteristic of the... more
Various kinds of distances of all negacyclic codes of length 2 s over 2 are completely determined. Using our structure theorems of negacyclic codes of length 2 s over 2 , we first calculate the Hamming distances of all such negacyclic... more
For any different odd primes ℓ and p, structure of constacyclic codes of length 2ℓ m p n over a finite field Fq of characteritic p and their duals is established in term of their generator polynomials. Among other results, all linear... more
We investigate negacyclic and cyclic codes of length p s over the finite field F p a. Negacyclic codes of length p s are precisely the ideals of the chain ring F p a [x] x p s +1. This structure is then used to obtain the Hamming distance... more
Iterative decoding of product codes over the binary erasure channel is investigated. It is shown that belief propagation decoding combined with cyclic permutations yields near-capacity performance with low complexity for high-code-rate... more
In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring R = F q + vF q + v 2 F q , where v 3 = v, for q odd. We give conditions on the existence of LCD codes and present... more
In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring $R=\F_{q}+v\F_{q}+v^{2}\F_{q}$, where $v^{3}=v$, for $q$ odd. We give conditions on the existence of LCD codes and... more
Let n be an even number such that n ≡ 0 (mod 4). We show that a power function x d , with d = 2 n+2 2 +2 n−2 2 −1, on F 2 n is an APN function of degree n/2 which is CCZ-equivalent to Kasami functions of degrees n/2 and (n + 4)/2.
Let F q be a finite field of order q, a prime power integer such that q = et + 1 where t ≥ 1, e ≥ 2 are integers. In this paper, we study cyclic codes of length n over a non-chain ring R e,q = F q [u]/ u e −1. We define a Gray map ϕ and... more
Linear complementary dual (LCD) codes and linear complementary pair (LCP) of codes over finite fields have been intensively studied recently due to their applications in cryptography, in the context of side channel and fault injection... more
For any finite commutative ring B with an identity there is a strict inclusion B[X ; Z 0 ] ⊂ B[X ; 1 2 ; Z 0 ] ⊂ B[X ; 1 2 2 Z 0 ] of commutative semigroup rings. This work is a continuation of Shah et al. (2011) [8], in which we extend... more
In his doctoral thesis, Snover proved that any binary $(m,256,\delta)$ code is equivalent to the Nordstrom-Robinson code or the punctured Nordstrom-Robinson code for $(m,\delta)=(16,6)$ or $(15,5)$ respectively. We prove that these codes... more
For a given binary BCH code Cn of length n = 2 s - 1 generated by a polynomial of degree r there is no binary BCH code of length (n + 1)n generated by a generalized polynomial of degree 2r. However, it does exist a binary cyclic code C... more
The problem of designing a fault-secure interface between a fault-tolerant RAM memory system and a transmission channel, both protected against errors using cyclic linear error detecting and/or correcting codes is considered. The main... more
The main objective of this research is to present the relationship between the subject of coding theory and three-dimensional projection space in the eighth field. We found the points, lines, and planes of the Galois field of order 8... more
A long-range fast-measurement Raman-based distributed temperature sensor (RDTS) with a meter spatial resolution exploiting a quasi-periodic pulse-coding technique is proposed in this paper. We experimentally demonstrate the feasibility of... more