7月16日：向青/岳勤/張耀祖/李崇道

Let $G$ be a finite abelian group. If $f: G\rightarrow {\bf C}$  is a nonzero function with Fourier transform $\hf$, the Donoho-Stark uncertainty principle states that $|\supp(f)||\supp(\hf)|\geq |G|$. The purpose of this talk 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, the sharpened uncertainty principle states, with notation above, that $$|\supp(f)||\supp(\hf)|\geq |G|+|\supp(f)|-|H(\supp(f))|,$$ where $H(\supp(f))$ is the stabilizer of $\supp(f)$ in $G$.

Let Fq be a finite field with q elements and p be a prime with gcd( p, q) = 1. Let G be a finite abelian p-group and Fq(G) be a group algebra. In this paper, we find all primitive idempotents and minimal abelian group codes in the group algebra Fq (G). Furthermore, we give all LCD abelian codes (linear code with complementary dual) and self-orthogonal abelian codes of Fq (G).

Quadratic residues (QR) codes, introduced by Prange in 1958, are cyclic codes with code rates not less than 1/2 and generally have large minimum distances, so that most of the known QR codes are the best-known codes. Both the famous Hamming code of length 7 and the Golay codes are QR codes. However, it is difficult to decode QR codes, and except for those of low lengths, the decoders for QR codes appeared quite late. The first algebraic decoder of Golay code of length 23 was proposed by Elia in 1987. From 1990, Reed et al. published a series of papers about algebraic decoding of QR codes of lengths 31, 41, 47, and 73. After that, the coding group of I-Shou University continued the QR decoding study and developed decoders of lengths 71, 79, 89, 97, 103, and 113. Hence, all binary QR codes of lengths not exceed 113 are decoded. In this presentation, we give a brief review of decoding QR codes. It also includes the most recent works done by I-Shou coding group which improve the decoding processes for the practical hardware implementation.

A Gaussian integer is a complex number whose real and imaginary parts are both integers. A Gaussian integer sequence is called \textit{perfect} if it satisfies the ideal periodic auto-correlation functions. That is, let $\mathbf S=(s(0),s(1),\ldots,s(N-1))$ be a complex sequence of period $N$, where $s(t)=u(t)+v(t)i$ for $u(t),v(t)\in\mathbb{Z}$, and $i=\sqrt{-1}$. The complex sequence $\mathbf S$ is said to be a {\em perfect Gaussian integer sequence} if \begin{eqnarray}
\label{Rsformula} R_{\mathbf S}(\tau)=\sum_{t=0}^{N-1}
s(t){s^*(t+\tau)}
\end{eqnarray}
is nonzero for $\tau=0$ and is zero for any $1\leq \tau \leq N-1$, where $s^*$ denotes the conjugate of a complex number $s$. The \textit{degree} of a Gaussian integer sequence is defined to be the number of distinct nonzero Gaussian integers within one period of the sequence. In fact, its minimum degree is two. This study proposes a new construction method, called monomial o-polynomials, to generate the minimum-degree perfect Gaussian integer sequences. The resulting sequences have odd periods and high energy efficiency. Furthermore, the number of cyclically distinct perfect Gaussian integer sequences is shown.

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