We study multidimensional linear complexity of periodic arrays over finite fields when the characteristic is coprime to every component of the period. We first relate the recurrence-ideal definition to an annihilator quotient in the group algebra and make the forward-shift convention explicit. Over a finite splitting field, we give a self-contained group-algebra derivation of the formula that identifies the complexity with the number of nonzero Fourier coefficients. For arrays obtained by the composition method, we give an exact spectral formula and a sufficient non-vanishing condition for the known upper bounds to be attained. In addition, we verify this condition for a linear shift rule and a Welch exponential shift rule. We then obtain closed formulas for composed arrays built from Legendre and quadratic character columns in odd characteristic.
Subspace coding is a fundamental technique for reliable communication in random linear network coding. In the multishot scenario, flag codes extend this structure by encoding information in sequences of nested subspaces. In this paper, we study the construction of optimum distance flag codes (ODFCs) via group actions. Starting from k-dimensional orbit subspace codes with maximum distance in Fqrk, for r≥2 we analyze how the associated group action can be lifted to the flag variety of a certain type vector. Our approach provides explicit orbital constructions of flag codes of a particular type vector whose cardinality attains the largest value allowed by known bounds. The resulting codes inherit the orbital structure of k-dimensional subspace codes, which simplifies the analysis of their size and distance. Moreover, the number of orbits required to attain the maximum cardinality is determined by the divisibility of qk−1 by r. Thus, our work generalizes the orbital construction of ODFCs in [4] under a certain restriction on type vector.
In this paper, we completely classify the non-trivial 2-(v,k,λ) designs with gcd(r,λ)=2, admitting an almost simple, flag-transitive automorphism group with the socle PSL(2,q).
This paper explores the structural interplay between binary matroids and binary linear codes, focusing on augmenting codes via splitting operations. For a binary linear code C with parity-check matrix H and minimum distance d, we established a sufficient condition for constructing a binary linear code C′ with improved minimum distance d′>d, via a splitting operation on the vector matroid M[H]. An increased minimum distance enhances a code's error-detecting and error-correcting capabilities, which is critical for improving reliability in data transmission and storage. We characterize a class of binary m-paving matroids whose generalized splitting matroid MT results in a m-paving matroids. We define a m-paving code as an (n,k) linear code C over GF(q), where 1<k<n−1, such that the matroid MG associated with its generator matrix G is a m-paving matroid. Using this definition, a class of binary m-paving codes for which the augmented dual results in a m-paving code is characterized. Additionally, we proved that if C is a binary self-dual code with generator matrix G, then the corresponding vector matroid M[G] is Eulerian.
Feedback shift registers, such as Linear Feedback Shift Registers (LFSRs), Multi-Recursive Matrix Methods (MRMMs), and Lagged Fibonacci Generators (LFGs), are fundamental components in stream cipher-based cryptographic systems. In this paper, we investigate systems composed of LFSRs under two distinct configurations. First, we study the cascade connection of LFSRs and demonstrate that it represents a special case of the first configuration. Under specific conditions, we derive the exact period of these cascaded systems. Second, we analyze a system comprising two LFSRs in the second configuration, where carry bits are introduced into the feedback computation of the second LFSR. We examine the periodicity of both the carry bits and the overall system. Furthermore, we generalize this construction to word size m, and show that an additive LFG can be represented by an equivalent system of LFSRs. This approach enables efficient LFG implementation in resource-constrained environments by using multiple LFSRs and a simple adder, thus eliminating the need for large word sizes.
