Ted Hurley

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 E<SUB align="right">jw, called 'measurements', are known for certain j ∈ J ⊂ {0, 1, . . . , m – 1} where {E<SUB align="right">i, 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(t<SUP align="right">2</SUP>). 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.]]>

Magnus Gausdal Find; Daniel Smith-Tone; Meltem SÃ¶nmez Turan

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.]]>

Anuradha Sharma; Amit K. Sharma

In this paper, we obtain Jacobi forms over k<SUB align="right">p from byte weight enumerators of self-dual codes over 𝔽<SUB align="right">p, where p is an odd prime, k<SUB align="right">p is the totally real field of the pth cyclotomic field and 𝔽<SUB align="right">p is the finite field of order p. We also determine Siegel modular forms of genus g (g ≥ 1 is an integer) over k<SUB align="right">p by substituting certain theta series into byte weight enumerators in genus g of self-dual codes over 𝔽<SUB align="right">p for all p ∈ 𝔓, where the set 𝔓 consists of all those odd primes p for which the ring of algebraic integers of k<SUB align="right">p 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.]]>

Sajjad Mazloum Panjehkeh; Gholam Reza Mohtashami Borzadaran; Mohammad Amini

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.]]>

Pierre-Louis Cayrel; Mbouye Khady Diagne; Cheikh ThiÃ©coumba Gueye

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).]]>

Amit Sharma; Maheshanand Bhaintwal

In this paper, we study a class of skew-constacyclic codes over R = ℤ<SUB align="right">4 + uℤ<SUB align="right">4, which is a non-chain extension of ℤ<SUB align="right">4. 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 ℤ<SUB align="right">4 have been found via this class, and the same have been added to the database of ℤ<SUB align="right">4 codes.]]>