Low-correlation-zone complementary sequence sets (LCZ-CSSs) play an important role in modern multi-carrier spread-spectrum and MIMO communication systems due to their favorable correlation characteristics within a prescribed delay range. In this paper, we propose a new class of periodic LCZ-CSSs with low inter-set cross-correlation properties by exploiting the combinatorial structure of circular quasi-Florentine rectangles. Furthermore, we show that the proposed LCZ-CSSs are asymptotically optimal with respect to the correlation bound derived by Liu et al. [IEEE Trans. Commun., 59(12) (2011), 3285-3289]. As an additional contribution, we present a new circular quasi-Florentine rectangle for N=10, which broadens the range of available circular quasi-Florentine rectangles for sequence design.
We consider the coding problem in the Stiefel manifold with chordal distance. After considering various low-dimensional instances of this problem, we use Rankin's bounds on spherical codes to prove upper bounds on the minimum distance of a Stiefel code, and then we construct several examples of codes that achieve equality in these bounds.
Linear codes with complementary duals (LCD codes) are pivotal in coding theory and cryptography due to their unique properties. Investigating subco des within well-known code classes is a recurring theme in coding theory, driven by diverse applications. This paper focuses on analyzing LCD subco des of LCD codes, with the main goal of deriving a closed mass formula for these subco des for any LCD code. Since the vector space Fnq over Fq is an LCD code of length n, the mass formula for LCD codes of length n, as established by Carlet et al. (IEEE TIT, 2019), can be viewed as a special case of counting the LCD subco des within this particular LCD code. Additionally, we explore the asymptotic distribution of specific subclasses of LCD subco des, providing further insights into their structural properties.
This article presents new families of quasi-cyclic binary codes (QC). The new QC codes are obtained based on the concept of a finite Euclidean plane and have the property of being self-orthogonal simultaneously to the Euclidean inner product and the symplectic inner product. The finite Euclidean plane is designed to be an "analogue" of the Euclidean plane, but the former is defined using the binary extension F2r of the field F2 = {0, 1}.
The Ipatov sequence is a ternary sequence known for its perfect periodic auto-correlation. Because of this property, Ipatov sequences are used in many communication and sensing systems. They are also part of the IEEE 802.15.4z standard. This paper investigates the aperiodic auto-correlation and periodic auto-ambiguity function of Ipatov sequences. Leveraging tools from exponential sums and quadratic forms over finite fields, we establish rigorous upper bounds on the magnitudes of these correlation properties.
Constructions of infinite families of almost MDS codes, Griesmer codes or quasi-perfect codes are challenging problems in coding theory. In this paper, we construct an infinite family of distance-optimal cyclic [q2m-1, q2m-3m-2, 4]q codes over Fq, where q is an arbitrary prime power. When m = 1, this family of almost MDS cyclic codes with Griesmer dual codes was constructed by Heng, Wang and Ding in their paper published in IEEE Trans. Inf. Theory, 2020. We prove that the covering radius of these almost MDS codes is three. Then we obtain many infinite families of their extended almost Griesmer and almost MDS [q2, q2-4, 4]q codes. Their dual codes are almost Griesmer or Griesmer. Moreover, we prove that any almost MDS code with the minimum distance 3 and the length n >= q + 2 is quasi-perfect and the almost MDS [q2 + 1, q2-3, 4]q code is quasi-perfect. This provides the first infinite family of almost MDS, almost Griesmer quasi-perfect codes with Griesmer dual codes.
In this paper, we discuss a family of linear evaluation codes over finite fields, called semigroup polytope codes, constructed from complements of difference hyperplane arrangements determined by numerical semigroups. The evaluation set is obtained from the finite-field complement of the arrangement associated with the canonical block decomposition of a numerical semigroup, yielding code lengths explicitly governed by the characteristic polynomial of the arrangement. This provides a new coding-theoretic framework that naturally links numerical semigroup invariants with finite-field geometry and combinatorial structures. We establish fundamental structural properties of these codes, including explicit formulas for the block length, dimension under injectivity hypotheses, and lower bounds for the minimum Hamming distance. The coordinate positions admit a natural Stirling-type stratification induced by block partitions, while the automorphism group of the associated block graph acts as a permutation automorphism group of the code, leading to orbit compression and symmetry-based reductions. We further compare these codes with Reed-Solomon, Reed-Muller, toric, and algebraic-geometry (AG)-Goppa codes, and discuss potential applications to reliable communication over noisy channels, network coding, code-based post-quantum cryptography and LRC structures.
MDS and NMDS codes are classical linear codes in coding theory that have been extensively studied for theoretical importance and wide applications. In this paper, we construct infinite families of 3-dimensional MDS and NMDS codes with length from q-1 to q + 2 (for prime q) by employing suitable generator matrices, and determine their weight enumerators. We further construct a class of NMDS codes with length q + 3 for odd q by adding suitable projective points to arcs in PG(2, q). Our results show that one class of MDS codes is non-GRS, and all constructed MDS codes are Griesmer codes whose duals are both length-optimal and dimension-optimal with respect to the Sphere packing bound. The NMDS codes obtained are near Griesmer codes, and their duals are nearly optimal under the same bound. In addition, we obtain several classes of optimal locally recoverable codes from the constructed NMDS codes.
In 2021, Jin et al. constructed four families of binary sequence sets via modified quadratic characters over finite fields. By analyzing rational points on elliptic curves and invoking the Hasse-Weil bound, they established that these sequence sets exhibit low correlation properties. In this paper, we propose an alternative approach: by transforming exponential sums involving quartic polynomials into those involving cubic polynomials and leveraging the more readily applicable Weil bound, we derive the same conclusions. Compared with their method, ours offers three key advantages: (i) it obviates the need for intricate analysis of elliptic curves, thereby streamlining technical procedures; (ii) it enables a more rigorous analysis of correlations; and (iii) it improves accessibility for researchers with limited expertise in algebraic geometry.
