On the Cross-Correlation of a p-Ary m-Sequence and Its Decimated Sequences by d=(pn+1)/(pk+1)+(pn-1)/2

Sung-Tai CHOI  Ji-Youp KIM  Jong-Seon NO  

IEICE TRANSACTIONS on Communications   Vol.E96-B   No.9   pp.2190-2197
Publication Date: 2013/09/01
Online ISSN: 1745-1345
DOI: 10.1587/transcom.E96.B.2190
Print ISSN: 0916-8516
Type of Manuscript: PAPER
Category: Fundamental Theories for Communications
cross-correlation,  cyclic code,  decimated sequence,  exponential sum,  m-sequence,  sequence family,  quadratic form,  weight distribution,  

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In this paper, for an odd prime p such that p≡3 mod 4, odd n, and d=(pn+1)/(pk+1)+(pn-1)/2 with k|n, the value distribution of the exponential sum S(a,b) is calculated as a and b run through $mathbb{F}_{p^n}$. The sequence family $mathcal{G}$ in which each sequence has the period of N=pn-1 is also constructed. The family size of $mathcal{G}$ is pn and the correlation magnitude is roughly upper bounded by $(p^k+1)sqrt{N}/2$. The weight distribution of the relevant cyclic code C over $mathbb{F}_p$ with the length N and the dimension ${ m dim}_{mathbb{F}_p}mathcal{C}=2n$ is also derived.