Construction of two- or three-weight binary linear codes from Vasil'ev codes
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
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.
Abstract
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.
참고문헌 (11)
- A. Ashikhmin, "Minimal vectors in linear codes", IEEE Transactions on Information Theory, 44(5), pp.2010~2017 (1998), Institute of Electrical and Electronics Engineers (IEEE)[DOI]
- R. Calderbank, "The geometry of two-weight codes", Bull. London Math. Soc, 18(2), pp.97~122 (1986)[DOI]
- Seunghwan Chang, "Linear codes from simplicial complexes", Designs, Codes and Cryptography, 86(10), pp.2167~2181 (2018), Springer Science and Business Media LLC[DOI]
- Yanling Chen, "A Lower Bound on the Optimum Distance Profiles of the Second-Order Reed–Muller Codes", IEEE Transactions on Information Theory, 56(9), pp.4309~4320 (2010), Institute of Electrical and Electronics Engineers (IEEE)[DOI]
- G. D. Cohen, "Cryptography and coding", pp.85~98 (2013), Springer
- Cunsheng Ding, "Minimal Binary Linear Codes", IEEE Transactions on Information Theory, 64(10), pp.6536~6545 (2018), Institute of Electrical and Electronics Engineers (IEEE)[DOI]
- C. Ding, "Covering and secret sharing with linear codes, in Discrete mathematics and theoretical computer science", pp.11~25 (2003), Springer
- T. Etzion, "Perfect binary codes: constructions, properties, and enumeration", IEEE Transactions on Information Theory, 40(3), pp.754~763 (1994), Institute of Electrical and Electronics Engineers (IEEE)[DOI]
- F. J. MacWilliams, "The Theory of Error-Correcting Codes. I" (1977), North-Holland Publishing Co
- Gary McGuire, "Quasi-Symmetric Designs and Codes Meeting the Grey–Rankin Bound", Journal of Combinatorial Theory, Series A, 78(2), pp.280~291 (1997), Elsevier BV[DOI]
- Ju. L. Vasil’ev, "On ungrouped, close-packed codes", Problemy Kibernet, (8), pp.337~339 (1962)