Two-weight and three-weight linear codes constructed from Weil sums

Abstract

<p style='text-indent:20px;'>Linear codes with few weights are widely used in strongly regular graphs, secret sharing schemes, association schemes and authentication codes. In this paper, we construct several two-weight and three-weight linear codes over finite fields by choosing suitable different defining sets. We also give some examples and some of the codes are optimal or almost optimal. Their applications to secret sharing schemes are also investigated.</p>

Linear Codes Constructed from Two Weakly Regular Plateaued Functions with Index (p − 1)/2

“…Linear codes with a few nonzero weights have attracted much attention in recent decades due to their wide applications in theory and practice, see [2][3][4][5][6][7][8][9][10][11]. Some linear codes are constructed from bent functions [6,12], square functions [13] and weakly regular plateaued functions [3,5,7].…”

“…For a codeword , its weight is defined by Then, the weight distribution of C is the sequence , where and stands for the number of codewords in C that have weight w , for , i.e., The code C is called t -weight if the number of nonzero for equals t . Linear codes with a few nonzero weights have attracted much attention in recent decades due to their wide applications in theory and practice, see [ 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 ]. Some linear codes are constructed from bent functions [ 6 , 12 ], square functions [ 13 ] and weakly regular plateaued functions [ 3 , 5 , 7 ].…”

Linear Codes From Two Weakly Regular Plateaued Functions with index (p-1)/2

“…The code C is called t-weight if the number of nonzero A w for 1 w n equals t. The weight distribution is of vital importance since it contains the information of computing the error probability of error detection and correction. In recent decades, a large number of linear codes have been investigated, most of which have a few weights and good parameters [3,4,7,8,10,12,16,17,22,23,25,26]. The construction of linear codes is usually based on different functions, such as, Boolean functions [3], bent functions [19,26], square functions [20] and weakly regular plateaued functions [4,16,17].…”