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: 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: 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: 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: 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: 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: 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

Abstract: In this work, we describe a double bordered construction of self-dual codes from group rings. We show that this construction is effective for groups of order 2p where p is odd, over the rings \(\mathbb {F}_{2}+u\mathbb {F}_{2}\) and \(\mathbb {F}_{4}+u\mathbb {F}_{4}\). We demonstrate the importance of this new construction by finding many new binary self-dual codes of lengths 64, 68 and 80; the new codes and their corresponding weight enumerators are listed in several tables. PubDate: 2020-01-09

Abstract: The maximum possible cardinality of a binary code of length n and Hamming distance d is denoted by A(n,d). The current lower bound for A(16,5) is 256, as implied by the Nordstrom–Robinson code. We improve this bound to 258 by presenting a binary code of length 16, minimum distance 5 and cardinality 258. The code is found using a known construction and Tabu Search. PubDate: 2020-01-01

Abstract: The second-order nonlinearity can provide knowledge on classes of Boolean functions used in symmetric-key cryptosystems, coding theory, and Gowers norm. It is well-known that bent functions possess the highest nonlinearity on even number of variables and so it will be of great interest to investigate the lower bound on the second-order nonlinearity of such functions. In 2008, Canteaut et al. (Finite Fields Appl. 14(1), 221–241, 2) found a class of monomial bent functions on n = 6r variables and proved that their derivatives have nonlinearities either 2n− 1 − 24r− 1 or 2n− 1 − 25r− 1. In this paper, we completely determine the distributions of the nonlinearities of the derivatives of this class of bent functions. Further, we present a new lower bound on the second-order nonlinearity of this class of bent functions, which is better than the previous one. PubDate: 2020-01-01

Abstract: In this paper, we study quasi-cyclic codes of index \(1\frac {1}{2}\) and co-index 2m over \(\mathbb {F}_{q}\) and their dual codes, where m is a positive integer, q is a power of an odd prime and \(\gcd (m,q) = 1\). We characterize and determine the algebraic structure and the minimal generating set of quasi-cyclic codes of index \(1\frac {1}{2}\) and co-index 2m over \(\mathbb {F}_{q}\). We note that some optimal and good linear codes over \(\mathbb {F}_{q}\) can be obtained from this class of codes. Furthermore, the algebraic structure of their dual codes is given. PubDate: 2020-01-01

Abstract: Double circulant codes of length 2n over the non-local ring \(R=\mathbb {F}_{q}+u\mathbb {F}_{q}, u^{2}=u,\) are studied when q is an odd prime power, and − 1 is a square in \(\mathbb {F}_{q}\). Double negacirculant codes of length 2n are studied over R when n is even, and q is an odd prime power. Exact enumeration of self-dual and LCD such codes for given length 2n is given. Employing a duality-preserving Gray map, self-dual and LCD codes of length 4n over \(\mathbb {F}_{q}\) are constructed. Using random coding and the Artin conjecture, the relative distance of these codes is bounded below for n →∞. The parameters of examples of modest lengths are computed. Several such codes are optimal. PubDate: 2020-01-01

Abstract: In 2003, Alfred Menezes, Edlyn Teske and Annegret Weng presented a conjecture on properties of the solutions of a type of quadratic equations over the binary extension fields, which had been confirmed by extensive experiments but the proof was unknown until now. We prove that this conjecture is correct. Furthermore, using this proved conjecture, we have completely determined the null space of a class of linearized polynomials. PubDate: 2020-01-01

Abstract: Let \(\mathbb {F}_{q}\) be the finite field of q elements and n = qm − 1 with m a positive integer. In this paper we construct a class of BCH and LCD BCH codes of length n over \(\mathbb {F}_{q}\) and investigate their dimensions and designed distance. Our results show that the designed distances of BCH and LCD BCH codes in this paper are larger than those in [11, Theorems 7, 10, 18, and 22]. It is viewed as a generalized result of [11]. PubDate: 2020-01-01

Abstract: In this paper, we introduce a new bordered construction for self-dual codes using group rings. We consider constructions over the binary field, the family of rings Rk and the ring \(\mathbb {F}_{4}+u\mathbb {F}_{4}\). We use groups of order 4, 12 and 20. We construct some extremal self-dual codes and non-extremal self-dual codes of length 16, 32, 48, 64 and 68. In particular, we construct 33 new extremal self-dual codes of length 68. PubDate: 2020-01-01

Abstract: Linear codes with complementary duals intersect with their duals trivially. Multinegacirculant codes that are complementary dual are characterized algebraically and some good codes are found in this family. Exact enumeration is performed for indices 2 and 3, whereas special choices of the co-index and base field size are needed for higher indices. Asymptotic existence results are derived for the special class of such codes that have co-index a power of two by means of Dickson polynomials. This shows that there are infinite families of complementary dual multinegacirculant codes with relative distance satisfying a modified Gilbert-Varshamov bound. PubDate: 2020-01-01

Abstract: Due to their simple construction, LFSRs are commonly used as building blocks in various random number generators. Nonlinear feedforward logic is incorporated in LFSRs to increase the linear complexity of the generated sequences. This work deals with Nonlinear Feedforward Generators (NLFGs) that generate sequences over arbitrary finite fields. We analyze the frequency of symbols in sequences generated by such configurations. Further, we propose a method of using nonlinear feedforward logic with word-based σ-LFSRs wherein vectors over a finite field are seen as elements of an extension field. We then briefly analyze sequences generated by an existing scheme and show that sequences generated by the proposed scheme are statistically more balanced. PubDate: 2020-01-01

Abstract: In this paper, we investigate all irreducible factors of \(x^{l_{1}^{m_{1}}l_{2}^{m_{2}}} - a\) over \(\mathbb {F}_{q}\) and obtain all primitive idempotents in \(\mathbb {F}_{q}[x]/\langle x^{l_{1}^{m_{1}}l_{2}^{m_{2}}} - a \rangle \), where \(a \in \mathbb {F}_{q}^{*}\), l1, l2 are two distinct odd prime divisors of qt − 1 with \(\gcd (l_{1}l_{2},q(q-1))= 1\) for prime t. Furthermore, the weight distributions of all irreducible constacyclic codes of length \(l_{1}^{m_{1}}l_{2}^{m_{2}}\) are presented for t = 2. As an application, we determine all linear complementary dual cyclic codes of length \(l_{1}^{m_{1}}l_{2}^{m_{2}}\) over \(\mathbb {F}_{q}\). PubDate: 2020-01-01

Abstract: In the present paper we introduce some sufficient conditions and a procedure for checking whether, for a given function, CCZ-equivalence is more general than EA-equivalence together with taking inverses of permutations. It is known from Budaghyan et al. (IEEE Trans. Inf. Theory 52.3, 1141–1152 2006; Finite Fields Appl. 15(2), 150–159 2009) that for quadratic APN functions (both monomial and polynomial cases) CCZ-equivalence is more general. We prove hereby that for non-quadratic APN functions CCZ-equivalence can be more general (by studying the only known APN function which is CCZ-inequivalent to both power functions and quadratics). On the contrary, we prove that for power non-Gold APN functions, CCZ equivalence coincides with EA-equivalence and inverse transformation for n ≤ 8. We conjecture that this is true for any n. PubDate: 2020-01-01

Abstract: Several classes of quaternary sequences of even period with optimal autocorrelation have been constructed by Su et al. based on interleaving certain kinds of binary sequences of odd period, i.e. Legendre sequence, twin-prime sequence and generalized GMW sequence. In this correspondence, the exact values of linear complexity over finite field \(\mathbb {F}_{4}\) and Galois ring \(\mathbb {Z}_{4}\) of the quaternary sequences are derived, respectively. PubDate: 2019-12-14