New Sets of Optimal Odd-length Binary Z-Complementary Pairs

A pair of sequences is called a Z-complementary pair (ZCP) if it has zero aperiodic autocorrelation sums (AACSs) for time-shifts within a certain region, called zero correlation zone (ZCZ). Optimal odd-length binary ZCPs (OB-ZCPs) display closest correlation properties to Golay complementary pairs (GCPs) in that each OB-ZCP achieves maximum ZCZ of width (N+1)/2 (where N is the sequence length) and every out-of-zone AACSs reaches the minimum magnitude value, i.e. 2. Till date, systematic constructions of optimal OB-ZCPs exist only for lengths $2^{\alpha} \pm 1$, where $\alpha$ is a positive integer. In this paper, we construct optimal OB-ZCPs of generic lengths $2^\alpha 10^\beta 26^\gamma +1$ (where $\alpha,~ \beta, ~ \gamma$ are non-negative integers and $\alpha \geq 1$) from inserted versions of binary GCPs. The key leading to the proposed constructions is several newly identified structure properties of binary GCPs obtained from Turyn's method. This key also allows us to construct OB-ZCPs with possible ZCZ widths of $4 \times 10^{\beta-1} +1$, $12 \times 26^{\gamma -1}+1$ and $12 \times 10^\beta 26^{\gamma -1}+1$ through proper insertions of GCPs of lengths $10^\beta,~ 26^\gamma, \text{and } 10^\beta 26^\gamma$, respectively. Our proposed OB-ZCPs have applications in communications and radar (as an alternative to GCPs)

New binary and ternary LCD codes

LCD codes are linear codes with important cryptographic applications. Recently, a method has been presented to transform any linear code into an LCD code with the same parameters when it is supported on a finite field with cardinality larger than 3. Hence, the study of LCD codes is mainly open for binary and ternary fields. Subfield-subcodes of $J$-affine variety codes are a generalization of BCH codes which have been successfully used for constructing good quantum codes. We describe binary and ternary LCD codes constructed as subfield-subcodes of $J$-affine variety codes and provide some new and good LCD codes coming from this construction

Deterministic Polynomial-Time Algorithms for Designing Short DNA Words

Designing short DNA words is a problem of constructing a set (i.e., code) of n DNA strings (i.e., words) with the minimum length such that the Hamming distance between each pair of words is at least k and the n words satisfy a set of additional constraints. This problem has applications in, e.g., DNA self-assembly and DNA arrays. Previous works include those that extended results from coding theory to obtain bounds on code and word sizes for biologically motivated constraints and those that applied heuristic local searches, genetic algorithms, and randomized algorithms. In particular, Kao, Sanghi, and Schweller (2009) developed polynomial-time randomized algorithms to construct n DNA words of length within a multiplicative constant of the smallest possible word length (e.g., 9 max{log n, k}) that satisfy various sets of constraints with high probability. In this paper, we give deterministic polynomial-time algorithms to construct DNA words based on derandomization techniques. Our algorithms can construct n DNA words of shorter length (e.g., 2.1 log n + 6.28 k) and satisfy the same sets of constraints as the words constructed by the algorithms of Kao et al. Furthermore, we extend these new algorithms to construct words that satisfy a larger set of constraints for which the algorithms of Kao et al. do not work.Comment: 27 page