Abstract: Abstract We correct some mistakes in the paper “A mass formula for negacyclic codes of length 2k and some good negacyclic codes over \(\mathbb {Z}_{4}+u\mathbb {Z}_{4}\)” (Bandi et al. Cryptogr. Commun. 9, 241–272, 2017). PubDate: 2020-03-25

Abstract: Abstract A bent4 function is a Boolean function with a flat spectrum with respect to a certain unitary transform \(\mathcal {T}\). It was shown previously that a Boolean function f in an even number of variables is bent4 if and only if f + σ is bent, where σ is a certain quadratic function depending on \(\mathcal {T}\). Hence bent4 functions are also called shifted bent functions. Similarly, a Boolean function f in an odd number of variables is bent4 if and only if f + σ is a semibent function satisfying some additional properties. In this article, for the first time, we analyse in detail the effect of the shifts on plateaued functions, on partially bent functions and on the linear structures of Boolean functions. We also discuss constructions of bent and bent4 functions from partially bent functions and study the differential properties of partially bent4 functions, unifying the previous work on partially bent functions. PubDate: 2020-03-23

Abstract: Abstract The concept of a locally repairable code (LRC) was introduced to protect the data from disk failures in large-scale storage systems. In this paper, we consider the LRCs with multiple disjoint repair sets and each repair set contains exactly one check symbol. By using several structures from combinatorial design theory, such as balanced incomplete block design, cyclic packing, group divisible design, near-Skolem sequence and Langford sequence, we construct several infinite classes of LRCs with the size of each repair set at most 3, which are optimal with respect to the bound proposed by Rawat et al. in 2016. PubDate: 2020-03-05

Abstract: Abstract Let \(R=\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle \). Then R is a local non-principal ideal ring of 16 elements. First, we give the structure of every cyclic code of odd length n over R and obtain a complete classification for these codes. Then we determine the cardinality, the type and its dual code for each of these cyclic codes. Moreover, we determine all self-dual cyclic codes of odd length n over R and provide a clear formula to count the number of these self-dual cyclic codes. Finally, we list some optimal 2-quasi-cyclic self-dual linear codes of length 30 over \(\mathbb {Z}_{4}\) and obtain 4-quasi-cyclic and formally self-dual binary linear [60,30,12] codes derived from cyclic codes of length 15 over \(\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle \). PubDate: 2020-03-01

Abstract: Abstract In this paper, we investigate a family of q2-ary narrow-sense and non-narrow-sense negacyclic BCH codes with length \(n=\frac {q^{2m}-1}{2}\), where q is an odd prime power and m ≥ 3 is odd. We propose Hermitian dual-containing conditions for narrow-sense and non-narrow-sense negacyclic BCH codes, and precisely compute the dimensions of these negacyclic BCH codes whose maximal designed distance can achieve \(\delta _{max}^{neg}\). Consequently, many new q-ary quantum codes can be derived from these dual-containing negacyclic BCH codes. Moreover, these new quantum codes are presented either with parameters better than or equal to the ones available in the literature, and also have larger designed distance than those from classical BCH codes. PubDate: 2020-03-01

Abstract: Abstract In this paper, we study the differential uniformity of the composition of two functions with the help of Boolean matrix theory. Based on the result of our research, we can construct new differentially 4-uniform permutations from known ones. In addition, we find some clues about the existence of APN permutations of \(\mathbb {F}_{2^{n}}\) for even n ≥ 8. PubDate: 2020-03-01

Abstract: Abstract Constacyclic codes are a subclass of linear codes and have been well studied. Constacyclic BCH codes are a family of constacyclic codes and contain BCH codes as a subclass. Compared with the in-depth study of BCH codes, there are relatively little study on constacyclic BCH codes. The objective of this paper is to determine the dimension and minimum distance of a class of q-ary constacyclic BCH codes of length \(\frac {q^{m}-1}{q-1}\) with designed distances \(\delta _{i}=q^{m-1}-\frac {q^{\lfloor \frac {m-3}2 \rfloor +i }-1}{q-1}\) for \(1\leq i\leq \min \limits \{\lceil \frac {m+1}2 \rceil -\lfloor \frac {m}{q+1} \rfloor , \lceil \frac {m-1}2 \rceil \}\). As will be seen, some of these codes are optimal. PubDate: 2020-03-01

Abstract: Abstract Let NFSR(f ) denote the nonlinear feedback shift register (NFSR) with characteristic function f = x0 ⊕ g(x1,x2,…,xn− 1) ⊕ xn. In this paper, the cycle structure of NFSR(fd) is discussed, where \(f^{d}=x_{0}\oplus g(x_{d},x_{2d},{\ldots } ,x_{(n-1)d})\oplus x_{nd}\) is also a characteristic function determined by f and a given integer d. If the cycle structure of NFSR(f ) is known, then it is shown that the cycle structure of NFSR(fd) can be completely determined. Moreover, with these results, three applications of the cycle structure of NFSR(fd) are presented: Firstly, the cycle structure of NFSR(fd) is discussed when f belongs to a class of symmetric characteristic functions. Compared with the previous work, our result can cover more cases while the proof is more straightforward. Secondly, we show the cycle structure of NFSR(fd) when f is a characteristic function of de Bruijn sequences and d = 2k. At last, a new necessary condition for f to be a characteristic function of de Bruijn sequences is presented, which can partially support the observation proposed in Çalik et al. (IEICE Trans. Fund. Electron. Commun. Comput. Sci. E93-A,(6), 1226–1231 2012) and Chan et al. (Lect. Notes Comput. Sci. 809, 166–173 1993). PubDate: 2020-03-01

Abstract: Abstract We construct a class of \(\mathbb {Z}_{p}\mathbb {Z}_{p^{s}}\)-additive cyclic codes generated by pairs of polynomials, where p is a prime number. The generator matrix of this class of codes is obtained. By establishing the relationship between the random \(\mathbb {Z}_{p}\mathbb {Z}_{p^{s}}\)-additive cyclic code and random quasi-cyclic code of index 2 over \(\mathbb {Z}_{p}\), the asymptotic properties of the rates and relative distances of this class of codes are studied. As a consequence, we prove that \(\mathbb {Z}_{p}\mathbb {Z}_{p^{s}}\)-additive cyclic codes are asymptotically good since the asymptotic GV-bound at \(\frac {1+p^{s-1}}{2}\delta \) is greater than \(\frac {1}{2}\), the relative distance of the code is convergent to δ, while the rate is convergent to \(\frac {1}{1+p^{s-1}}\) for \(0< \delta < \frac {1}{1+p^{s-1}}\). PubDate: 2020-03-01

Abstract: Abstract This paper presents a class of integer codes capable of correcting l-bit burst asymmetric errors within a b-bit byte (1 ≤ l < b) and double asymmetric errors within a codeword. The presented codes are constructed with the help of a computer and have the potential to be used in unamplified optical networks. In addition, the paper derives the upper bound on code length and shows that the proposed codes are efficient in terms of redundancy. PubDate: 2020-03-01

Abstract: Abstract Linear complementary dual codes (or codes with complementary duals) are codes whose intersections with their dual codes are trivial. We study the largest minimum weights d(n,k) among all binary linear complementary dual [n,k] codes. We determine d(n,4) for n ≡ 2,3,4,5,6,9,10,13 (mod 15), and d(n,5) for n ≡ 3,4,5,7,11,19,20, 22,26 (mod 31). Combined with known results, d(n,k) are also determined for n ≤ 24. PubDate: 2020-03-01

Abstract: Abstract Involutions over finite fields are permutations whose compositional inverses are themselves. Involutions especially over \( \mathbb {F}_{q} \) with q is even have been used in many applications, including cryptography and coding theory. The explicit study of involutions (including their fixed points) has started with the paper (Charpin et al. IEEE Trans. Inf. Theory, 62(4), 2266–2276 2016) for binary fields and since then a lot of attention had been made in this direction following it; see for example, Charpin et al. (2016), Coulter and Mesnager (IEEE Trans. Inf. Theory, 64(4), 2979–2986, 2018), Fu and Feng (2017), Wang (Finite Fields Appl., 45, 422–427, 2017) and Zheng et al. (2019). In this paper, we study constructions of involutions over finite fields by proposing an involutory version of the AGW Criterion. We demonstrate our general construction method by considering polynomials of different forms. First, in the multiplicative case, we present some necessary conditions of f(x) = xrh(xs) over \(\mathbb {F}_{q}\) to be involutory on \(\mathbb {F}_{q}\), where s∣(q − 1). Based on this, we provide three explicit classes of involutions of the form xrh(xq− 1) over \(\mathbb {F}_{q^{2}}\). Recently, Zheng et al. (Finite Fields Appl., 56, 1–16 2019) found an equivalent relationship between permutation polynomials of \(g(x)^{q^{i}} - g(x) + cx +(1-c)\delta \) and \(g\left (x^{q^{i}} - x + \delta \right ) +c x\). The other part work of this paper is to consider the involutory property of these two classes of permutation polynomials, which fall into the additive case of the AGW criterion. On one hand, we reveal the relationship of being involutory between the form \( g(x)^{q^{i}} - g(x) + cx +(1-c)\delta \) and the form \( g\left (x^{q^{i}} - x + \delta \right ) +c x \) over \( \mathbb {F}_{q^{m}} \) ; on the other hand, the compositional inverses of permutation polynomials of the form \( g\left (x^{q^{i}} - x + \delta \right ) + cx \) over \( \mathbb {F}_{q^{m}} \) are computed, where \( \delta \in \mathbb {F}_{q^{m}} \), \( g(x) \in \mathbb {F}_{q^{m}}[x] \) and integers m, i satisfy 1 ≤ i ≤ m − 1. In addition, a class of involutions of the form \( g\left (x^{q^{i}} - x + \delta \right ) + cx \) is constructed. Finally, we study the fixed points of constructed involutions and compute the number of all involutions with any given number of fixed points over \( \mathbb {F}_{q} \). PubDate: 2020-03-01

Abstract: Abstract In this paper we study almost p-ary sequences and their autocorrelation coefficients. We first study the number ℓ of distinct out-of-phase autocorrelation coefficients for an almost p-ary sequence of period n + s with s consecutive zero-symbols. We prove an upper bound and a lower bound on ℓ. It is shown that ℓ can not be less than \(\min \limits \{s,p,n\}\). In particular, it is shown that a nearly perfect sequence with at least two consecutive zero symbols does not exist. Next we define a new difference set, partial direct product difference set (PDPDS), and we prove the connection between an almost p-ary nearly perfect sequence of type (γ1, γ2) and period n + 2 with two consecutive zero-symbols and a cyclic \((n+2,p,n,\frac {n-\gamma _{2} - 2}{p}+\gamma _{2},0,\frac {n-\gamma _{1} -1}{p}+\gamma _{1},\frac {n-\gamma _{2} - 2}{p},\frac {n-\gamma _{1} -1}{p})\) PDPDS for arbitrary integers γ1 and γ2. Then we prove a necessary condition on γ2 for the existence of such sequences. In particular, we show that they do not exist for γ2 ≤ − 3. PubDate: 2020-02-17

Abstract: Abstract This paper presents an explicit representation for the solutions of the equation \({\sum }_{i=0}^{\frac kl-1}x^{2^{li}} = a \in \mathbb {F}_{2^{n}}\) for any given positive integers k, l with l k and n, in the closed field \({\overline {\mathbb {F}_{2}}}\) and in the finite field \(\mathbb {F}_{2^{n}}\). As a by-product of our study, we are able to completely characterize the a’s for which this equation has solutions in \(\mathbb {F}_{2^{n}}\). PubDate: 2020-02-11

Abstract: Abstract We consider a special type of sequence of arithmetic progressions, in which consecutive progressions are related by the property: ithterms ofjth, (j + 1)thprogressions of the sequence are multiplicative inverses of each other modulo(i + 1)thterm ofjthprogression. Such a sequence is uniquely defined for any pair of co-prime numbers. A computational problem, defined in the context of such a sequence and its generalization, is shown to be equivalent to the integer factoring problem. The proof is probabilistic. As an application of the equivalence result, we propose a method for how users securely agree upon secret keys, which are ensured to be random. We compare our method with factoring based public key cryptographic systems: RSA (Rivest et al., ACM 21, 120–126, 1978) and Rabin systems (Rabin 1978). We discuss the advantages of the method, and its potential use-case in the post quantum scenario. PubDate: 2020-01-30

Abstract: Abstract In 2018, Ding et al. introduced a new generalisation of the punctured binary Reed-Muller codes to construct LCD codes and 2-designs. They studied the minimum distance of the codes and proposed an open problem about the minimum distance. In this paper, several new results on the minimum distance of the generalised punctured binary Reed- Muller are presented. Particularly, some of the results are a generalisation or improvement of previous results in (Finite Fields Appl. 53, 144–174, 2018). PubDate: 2020-01-11

Abstract: Abstract We present new families of three-dimensional (3-D) optical orthogonal codes for applications to optical code-division multiple access (OCDMA) networks. The families are based in the extended rational cycle used for the 2-D Moreno-Maric construction. The new families are asymptotically optimal with respect to the Johnson bound. PubDate: 2020-01-10

Abstract: Abstract In this paper, we show that LCD codes are not equivalent to non-LCD codes over small finite fields. The enumeration of binary optimal LCD codes is obtained. We also get the exact value of LD(n,2) over \(\mathbb {F}_{3}\) and \(\mathbb {F}_{4}\), where LD(n,2) := max{d∣thereexsitsan [n,2, d] LCD\( code~ over~ \mathbb {F}_{q}\}\). We study the bound of LCD codes over \(\mathbb {F}_{q}\) and generalize a conjecture proposed by Galvez et al. about the minimum distance of binary LCD codes. PubDate: 2020-01-09

Abstract: Abstract We introduce almost supplementary difference sets (ASDS). For odd m, certain ASDS in \(\mathbb Z_{m}\) that have amicable incidence matrices are equivalent to quaternary sequences of odd length m with optimal autocorrelation. As one consequence, if 2m − 1 is a prime power, or m ≡ 1 mod 4 is prime, then ASDS of this kind exist. We also explore connections to optimal binary sequences and group cohomology. PubDate: 2020-01-09