Journal of the
Korean Mathematical Society JKMS

The set $D$ of column vectors of a generator matrix of a linear code is called a defining set of the linear code. In this paper we consider the problem of constructing few-weight (mainly two- or three-weight) linear codes from defining sets. It can be easily seen that we obtain an one-weight code when we take a defining set to be the nonzero codewords of a linear code. Therefore we have to choose a defining set from a non-linear code to obtain two- or three-weight codes, and we face the problem that the constructed code contains many weights. To overcome this difficulty, we employ the linear codes of the following form: Let $D$ be a subset of $\mathbb{F}_2^n$, and $W$ (resp.~$V$) be a subspace of $\mathbb{F}_2$ (resp.~$\mathbb{F}_2^n$). We define the linear code $\mathcal{C}_D(W; V)$ with defining set $D$ and restricted to $W, V$ by \[ \mathcal{C}_D(W; V) = \{(s+u\cdot x)_{x\in D^*} \,|\, s\in W, u\in V\}. \] We obtain two- or three-weight codes by taking $D$ to be a Vasil'ev code of length $n=2^m-1 (m \geq 3)$ and a suitable choices of $W$. We do the same job for $D$ being the complement of a Vasil'ev code. The constructed few-weight codes share some nice properties. Some of them are optimal in the sense that they attain either the Griesmer bound or the Grey-Rankin bound. Most of them are minimal codes which, in turn, have an application in secret sharing schemes. Finally we obtain an infinite family of minimal codes for which the sufficient condition of Ashikhmin and Barg does not hold.

Keywords: Vasil'ev code, Grey-Rankin bound, minimal linear codes, few-weight codes

MSC numbers: 94B05

Supported by: The first author was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (2014R1A1A2A10054745)