In this paper, we study a class of linear codes defined by characteristic functions of certain subsets of a finite field. We derive a sufficient and necessary condition for such a code to be a minimal linear code by a character-theoretical approach. We obtain new three-weight or four-weight minimal linear codes that do not satisfy the Ashikhmin-Barg condition by using partial difference sets. We show that our construction yields minimal linear codes that do not arise from cutting vectorial blocking sets, and also discuss their applications in secret sharing schemes.

Let G be a finite abelian group. If f : G → C is a nonzero function with Fourier transform f, the Donoho-Stark uncertainty principle states that |supp(f)||supp(f̂)| ≥ |G|. The purpose of this paper is twofold. First, we present the shift bound for abelian codes with a streamlined proof. Second, we use the shifting technique to prove a generalization and a sharpening of the Donoho-Stark uncertainty principle. In particular, if f : G → F is a non-zero function from G to a field F, and if f has a Fourier transform f̂, the sharpened uncertainty principle states that |supp(f)||supp(f̂)| ≥ |G|+|supp(f)|-|H(supp(f))|, where H(supp(f)) is the stabilizer of supp(f) in G.

We give necessary conditions for a two-weight projective cyclic code to be the direct sum of two one-weight irreducible cyclic subcodes of the same dimension, following the work of Wolfmann and Vega. This confirms Vega's conjecture that all the two-weight cyclic codes of this type are the known ones in the projective case.

Recently, there has been intensive research on the weight distributions of cyclic codes. In this paper, we compute the weight distributions of three classes of cyclic codes with Niho exponents. More specifically, we obtain two classes of binary three-weight and four-weight cyclic codes and a class of nonbinary four-weight cyclic codes. The weight distributions follow from the determination of value distributions of certain exponential sums. Several examples are presented to show that some of our codes are optimal and some have the best known parameters.

The determination of the cross correlation between an m-sequence and its decimated sequence has been a longstanding research problem. Considering a ternary m-sequence of period 33r- 1, we determine the cross correlation distribution for decimations d = 3r+ 2 and d = 32r+ 2, where gcd(r, 3) = 1. Meanwhile, for a binary m-sequence of period 22lm- 1, we make an initial investigation for the decimation d = (22lm- 1)/(2m+ 1) + 2s, where l ≥ 2 is even and 0 <; s <; 2m - 1. It is shown that the cross correlation takes at least four values. Furthermore, we confirm the validity of two famous conjectures due to Sarwate et al. and Helleseth in this case.

The duals of cyclic codes with two zeros have been extensively studied, and their weight distributions have recently been evaluated in some cases. In this paper, we determine the weight distribution of a certain new class of such codes by computations involving index 2 Gauss sums.

The determination of weight distribution of cyclic codes involves the evaluation of Gauss sums and exponential sums. Despite some cases where a neat expression is available, the computation is generally rather complicated. In this note, we determine the weight distribution of a class of reducible cyclic codes whose dual codes may have arbitrarily many zeros. This goal is achieved by building an unexpected connection between the corresponding exponential sums and the spectra of Hermitian forms graphs.

Codebooks (also called signal sets) meeting the Welch bound on the maximum correlation amplitude are called MWBE codebooks and are desirable in code-division multiple-access systems. Two different but related constructions of MWBE codebooks from difference sets were developed by Xia and Ding recently. The objectives of this correspondence are to present a generic construction of MWBE codebooks that contains the previous two constructions as special cases and describe new MWBE codebooks that cannot be produced by the earlier two constructions.