# Forthcoming articles

International Journal of Information and Coding Theory

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 International Journal of Information and Coding Theory (8 papers in press)  Regular Issues  $\mathbb{Z}_p\mathbb{Z}_p[u]$-Additive cyclic codes   by Lingyu Diao, Jian Gao Abstract: Additive cyclic codes of length $(\alpha,\beta)$ over $\mathbb{Z}_p\mathbb{Z}_p[u]$ can be viewed as $\mathbb{Z}_p[u][x]$-submodules of $\mathbb{Z}_p[x]/(x^\alpha-1)\times \mathbb{Z}_p[u][x]/(x^\beta-1)$, where $\mathbb{Z}_p[u]=\mathbb{Z}_p+u\mathbb{Z}_p$, $u^2=0$. In this paper, we determine the generator polynomials and the minimal generating sets of this family of codes as $\mathbb{Z}_p[u]$-submodules of $\mathbb{Z}_p[x]/(x^\alpha-1)\times \mathbb{Z}_p[u][x]/(x^\beta-1)$. Further, we also determine the generator polynomials of the dual codes of $\mathbb{Z}_p\mathbb{Z}_p[u]$-additive cyclic codes. Moreover, some binary quantum codes are constructed by additive cyclic codes over $\mathbb{Z}_2\mathbb{Z}_2[u]$. Keywords: additive cyclic codes; minimal generating sets; binary quantum codes. A deterministic algorithm for the distance and weight distribution of binary nonlinear codes   by Emanuele Bellini, Massimiliano Sala Abstract: Given a binary nonlinear code, we provide a deterministic algorithm to compute its weight and distance distribution, rnand in particular its minimum weight and its minimum distance,rnwhich takes advantage of fast Fourier techniques.rnThis algorithm's performance is similar to that of best-known algorithms for the average case, rnwhile it is especially efficient for codes with low information rate. rnWe provide complexity estimates for several cases of interest. Keywords: Distance distribution; minimum distance; weight distribution; minimum weight; nonlinear code. Solving underdetermined systems with error-correcting codes   by Ted Hurley Abstract: In an underdetermined system of equations Aw = y, where A is an m × n matrix, only u of the entries of y with u < m are known. Thus Ejw, called 'measurements', are known for certain j ∈ J ⊂ {0, 1, . . . , m – 1} where {Ei, i = 0, 1, . . . , m – 1} are the rows of A and |J| = u. It is required, if possible, to solve the system uniquely when x has at most t non-zero entries with u ≥ 2t. Here such systems are considered from an error-correcting coding point of view. This reduces the problem to finding a suitable decoding algorithm which then finds w. Decoding workable algorithms are shown to exist, from which the unknown w may be determined, in cases where the known 2t values are evenly spaced, that is when the elements of J are in arithmetic sequence. The method can be applied to Fourier matrices and certain classes of Vandermonde matrices. The decoding algorithm has complexity O(nt) and in some cases the complexity is O(t2). Matrices which have the property that the determinant of any square submatrix is non-zero are of particular interest. Randomly choosing rows of such matrices can then give t error-correcting pairs to generate a 'measuring' code. This has applications to signal processing and compressed sensing. Keywords: codes; compressed sensing; signal processing; underdetermined system.DOI: 10.1504/IJICOT.2017.10005825  The number of boolean functions with multiplicative complexity 2   by Magnus Gausdal Find, Daniel Smith-Tone, Meltem Sönmez Turan Abstract: Multiplicative complexity is a complexity measure defined as the minimum number of AND gates required to implement a given primitive by a circuit over the basis (AND, XOR, NOT). Implementations of cyphers with a small number of AND gates are preferred in protocols for fully homomorphic encryption, multiparty computation and zero-knowledge proofs. Fischer and Peralta (2002) computed the number of n-variable Boolean functions with multiplicative complexity 1. In this paper, we study Boolean functions that can be constructed with two AND gates. By characterising the structure of these functions in terms of affine equivalence relations, we provide a closed-form formula for the number of Boolean functions with multiplicative complexity 2. Keywords: affine transformations; Boolean circuits; circuit complexity; cryptography.DOI: 10.1504/IJICOT.2017.10005826  Byte weight enumerators of codes over 𝔽p and modular forms over a totally real field   by Anuradha Sharma, Amit K. Sharma Abstract: In this paper, we obtain Jacobi forms over kp from byte weight enumerators of self-dual codes over 𝔽p, where p is an odd prime, kp is the totally real field of the pth cyclotomic field and 𝔽p is the finite field of order p. We also determine Siegel modular forms of genus g (g ≥ 1 is an integer) over kp by substituting certain theta series into byte weight enumerators in genus g of self-dual codes over 𝔽p for all p ∈ 𝔓, where the set 𝔓 consists of all those odd primes p for which the ring of algebraic integers of kp is a Euclidean domain. Further, we define some partial Epstein zeta functions and derive their functional equation using the Mellin transform of the theta series. Keywords: algebraic integers; lattices; theta series.DOI: 10.1504/IJICOT.2017.10005828  Results related to exponential entropy   by Sajjad Mazloum Panjehkeh, Gholam Reza Mohtashami Borzadaran, Mohammad Amini Abstract: The Shannon entropy which is widely used in many fields of science, some advantage and disadvantage of it are noticeable. In view of disadvantage formulated, other entropies like exponential entropy. In this paper, we study some characterisation and properties of exponential entropy such as asymptotic equipartition property, chain rule, the subadditivity, invariancy under monotone transformation in parallel to those for the Shannon entropy in continuous and discrete cases. Also, the relationship between exponential entropy with Tsallis entropy and Renyi entropy obtained. Finally, we show that in the image segmentation, algorithms based on exponential entropy has a better performance than the algorithms which are based on the Shannon entropy. Keywords: chain rule; exponential entropy; information theory; jensen's inequality; Shannon entropy; Tsallis entropy.DOI: 10.1504/IJICOT.2017.10005832  NP-completeness of the Goppa parameterised random binary quasi-dyadic syndrome decoding problem   by Pierre-Louis Cayrel, Mbouye Khady Diagne, Cheikh Thiécoumba Gueye Abstract: In 1978, the syndrome decoding problem (SDP) was proven to be NP-complete for random binary codes. Since then, the security of several cryptographic applications relies on its hardness. In 2009, Finiasz extended this result by demonstrating the NP-completeness of certain subclasses of SDP. In this paper, we prove the NP-completeness of the Goppa parameterised quasi-dyadic syndrome decoding problem. We use a reduction to the four-dimensional matching problem (proven NP-complete). Keywords: four dimensional matching problem; NP-complete; quasi-dyadic Goppa codes; syndrome decoding problem.DOI: 10.1504/IJICOT.2017.10005835  A class of skew-constacyclic codes over ℤ4 + uℤ4   by Amit Sharma, Maheshanand Bhaintwal Abstract: In this paper, we study a class of skew-constacyclic codes over R = ℤ4 + uℤ4, which is a non-chain extension of ℤ4. Some structural properties of R[x, θ] are discussed, where θ is an automorphism of R. We determine a necessary condition and a sufficient condition for these codes to be free, when they are principally generated. A Gray map over R is defined and some good codes are obtained using it. For even n, a relation between the generator polynomial of a code and that of its dual is obtained. Some examples are given to illustrate the results. Further, we have generalised these codes to double skew-constacyclic codes over R. Some good codes with improved minimum Lee distance over ℤ4 have been found via this class, and the same have been added to the database of ℤ4 codes. Keywords: Codes over ℤ4 + uℤ4; constacyclic codes; factorisations of polynomials; Gray map; skew polynomial rings.DOI: 10.1504/IJICOT.2017.10005836