2024/01/15 by Cherubini, Giacomo, Micheli, Giacomo
#11T06 #11T71 #68P30 #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2401.07986
Let n be a prime power, r be a prime with r| n-1, and ε∈ (0,1/2). Using the theory of multiplicative character sums and superelliptic curves, we construct new codes over \mathbb Fr having length n, relative distance (r-1)/r+O(n-ε) and rate n-1/2-ε. When r=2, our binary codes have exponential size when compared to all previously known families of linear and non-linear codes with relative distance asymptotic to 1/2, such as Delsarte--Goethals codes. Moreover, concatenating with a Reed--Solomon code gives a family of codes of length n, asymptotic distance 1/2 and rate Ω(n-ε) for any fixed small ε>0, improving our initial construction. Such rate is also asymptotically better than the one by Kschischang and Tasbihi obtained by concatenating a Reed--Solomon with Reed--Muller, improving by a factor in Ω(n1/2/log(n)).