# American Institute of Mathematical Sciences

February  2019, 13(1): 195-211. doi: 10.3934/amc.2019013

## Some two-weight and three-weight linear codes

 1 Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200062, China 2 Department of Mathematics, KAIST, Daejeon, 305-701, Korea 3 School of Mathematical Sciences, Qufu Normal University, Shandong 273165, China

* Corresponding author: Chengju Li

Received  August 2018 Published  December 2018

Fund Project: Chengju Li was supported by the National Natural Science Foundation of China under Grant 11701179, the Shanghai Sailing Program under Grant 17YF1404300, and the Foundation of Science and Technology on Information Assurance Laboratory under Grant KJ-17-007.
Shudi Yang was supported by the National Natural Science Foundation of China under Grants 11701317 and 11431015, China Postdoctoral Science Foundation Funded Project under Grant 2017M611801, and Jiangsu Planned Projects for Postdoctoral Research Funds under Grant 1701104C.

Let
 $\Bbb F_q$
be the finite field with
 $q = p^m$
elements, where
 $p$
is an odd prime and
 $m$
is a positive integer. For a positive integer
 $t$
, let
 $D \subset \Bbb F_q^t$
and let
 $\mbox{Tr}_m$
be the trace function from
 $\Bbb F_q$
onto
 $\Bbb F_p$
. We define a
 $p$
-ary linear code
 $\mathcal C_D$
by
 $\mathcal C_D = \{\textbf{c}(a_1,a_2, ..., a_t): a_1, a_2, ..., a_t ∈ \Bbb F_{p^m}\},$
where
 $\textbf{c}(a_1,a_2, ..., a_t) = \big(\mbox{Tr}_m(a_1x_1+a_2x_2+···+a_tx_t)\big)_{(x_1,x_2, ..., x_t)∈ D}.$
In this paper, we will present the weight enumerators of the linear codes
 $\mathcal C_D$
in the following two cases:
1.
 $D = \{(x_1,x_2, ..., x_t) ∈ \Bbb F_q^t \setminus \{(0,0, ..., 0)\}: \mbox{Tr}_m(x_1^2+x_2^2+···+x_t^2) = 0\}$
;
2.
 $D = \{(x_1,x_2, ..., x_t) ∈ \Bbb F_q^t: \mbox{Tr}_m(x_1^2+x_2^2+···+x_t^2) = 1\}$
.
It is shown that
 $\mathcal C_D$
is a two-weight code if
 $tm$
is even and three-weight code if
 $tm$
is odd in both cases. The weight enumerators of
 $\mathcal C_D$
in the first case generalize the results in [17] and [18]. The complete weight enumerators of
 $\mathcal C_D$
are also investigated.
Citation: Chengju Li, Sunghan Bae, Shudi Yang. Some two-weight and three-weight linear codes. Advances in Mathematics of Communications, 2019, 13 (1) : 195-211. doi: 10.3934/amc.2019013
##### References:

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##### References:
Weight enumerators of Theorem 3.2 for odd $tm$
 Weight Frequency 0 1 $(p-1)p^{tm-2}$ $p^{tm-1}-1$ $(p-1)(p^{tm-2}-p^{\frac {tm-3} 2})$ $\frac {p-1} 2(p^{tm-1}+p^{\frac {tm-1} 2})$ $(p-1)(p^{tm-2}+p^{\frac {tm-3} 2})$ $\frac {p-1} 2(p^{tm-1}-p^{\frac {tm-1} 2})$
 Weight Frequency 0 1 $(p-1)p^{tm-2}$ $p^{tm-1}-1$ $(p-1)(p^{tm-2}-p^{\frac {tm-3} 2})$ $\frac {p-1} 2(p^{tm-1}+p^{\frac {tm-1} 2})$ $(p-1)(p^{tm-2}+p^{\frac {tm-3} 2})$ $\frac {p-1} 2(p^{tm-1}-p^{\frac {tm-1} 2})$
Weight enumerators of Theorem 3.2 for even $tm$
 Weight Frequency 0 1 $(p-1)p^{tm-2}$ $p^{tm-1}+(-1)^{(\frac {m(p-1)} 4+1)t}(p-1)p^{\frac {tm-2} 2}-1$ $(p-1)\big(p^{tm-2}+(-1)^{(\frac {m(p-1)} 4+1)t}p^{\frac {tm-2} 2}\big)$ $(p-1)\big(p^{tm-1}-(-1)^{(\frac {m(p-1)} 4+1)t}p^{\frac {tm-2} 2}\big)$
 Weight Frequency 0 1 $(p-1)p^{tm-2}$ $p^{tm-1}+(-1)^{(\frac {m(p-1)} 4+1)t}(p-1)p^{\frac {tm-2} 2}-1$ $(p-1)\big(p^{tm-2}+(-1)^{(\frac {m(p-1)} 4+1)t}p^{\frac {tm-2} 2}\big)$ $(p-1)\big(p^{tm-1}-(-1)^{(\frac {m(p-1)} 4+1)t}p^{\frac {tm-2} 2}\big)$
Weight enumerators of Theorem 4.1 for odd $tm$
 Weight Frequency 0 1 $(p-1)p^{tm-2}$ $p^{tm-1}-1$ $(p-1)p^{tm-2}+(-1)^{\frac {(p-1)(tm+3)} 4}p^{\frac {tm-1} 2}+p^{\frac {tm-3} 2}$ $\frac {p-1} 2(p^{tm-1}+p^{\frac {tm-1} 2})$ $(p-1)p^{tm-2}+(-1)^{\frac {(p-1)(tm+3)} 4}p^{\frac {tm-1} 2}-p^{\frac {tm-3} 2}$ $\frac {p-1} 2(p^{tm-1}-p^{\frac {tm-1} 2})$
 Weight Frequency 0 1 $(p-1)p^{tm-2}$ $p^{tm-1}-1$ $(p-1)p^{tm-2}+(-1)^{\frac {(p-1)(tm+3)} 4}p^{\frac {tm-1} 2}+p^{\frac {tm-3} 2}$ $\frac {p-1} 2(p^{tm-1}+p^{\frac {tm-1} 2})$ $(p-1)p^{tm-2}+(-1)^{\frac {(p-1)(tm+3)} 4}p^{\frac {tm-1} 2}-p^{\frac {tm-3} 2}$ $\frac {p-1} 2(p^{tm-1}-p^{\frac {tm-1} 2})$
Weight enumerators of Theorem 4.1 for even $tm$
 $2 \nmid \big(\frac {m(p-1)} 4+1\big)t$ Weight Frequency 0 1 $(p-1)p^{tm-2}$ $\frac {p+1} 2p^{tm-1}-\frac {p-1} 2 p^{\frac {tm-2} 2}-1$ $(p-1)p^{tm-2}+2p^{\frac {tm-2} 2}$ $\frac {p-1} 2\big(p^{tm-1}+p^{\frac {tm-2} 2}\big)$
 $2 \nmid \big(\frac {m(p-1)} 4+1\big)t$ Weight Frequency 0 1 $(p-1)p^{tm-2}$ $\frac {p+1} 2p^{tm-1}-\frac {p-1} 2 p^{\frac {tm-2} 2}-1$ $(p-1)p^{tm-2}+2p^{\frac {tm-2} 2}$ $\frac {p-1} 2\big(p^{tm-1}+p^{\frac {tm-2} 2}\big)$
Weight enumerators of Theorem 4.1 for even $tm$
 $2 \mid \big(\frac {m(p-1)} 4+1\big)t$ Weight Frequency 0 1 $(p-1)p^{tm-2}$ $\frac {p+1} 2p^{tm-1}+\frac {p-1} 2 p^{\frac {tm-2} 2}-1$ $(p-1)p^{tm-2}-2p^{\frac {tm-2} 2}$ $\frac {p-1} 2\big(p^{tm-1}-p^{\frac {tm-2} 2}\big)$
 $2 \mid \big(\frac {m(p-1)} 4+1\big)t$ Weight Frequency 0 1 $(p-1)p^{tm-2}$ $\frac {p+1} 2p^{tm-1}+\frac {p-1} 2 p^{\frac {tm-2} 2}-1$ $(p-1)p^{tm-2}-2p^{\frac {tm-2} 2}$ $\frac {p-1} 2\big(p^{tm-1}-p^{\frac {tm-2} 2}\big)$
Complete weight enumerators of Theorem 5.1 for odd $tm$
 $N_0=n-\sum_{\rho \in \Bbb F_p^*}N_\rho$ $N_\rho (\rho \in \Bbb F_p^*)$ Frequency 0 1 $p^{tm-2}$ $p^{tm-1}-1$ $p^{tm-2}-p^{\frac {tm-3} 2}$ $\frac {p-1} 2(p^{tm-1}+p^{\frac {tm-1} 2})$ $p^{tm-2}+p^{\frac {tm-3} 2}$ $\frac {p-1} 2(p^{tm-1}-p^{\frac {tm-1} 2})$
 $N_0=n-\sum_{\rho \in \Bbb F_p^*}N_\rho$ $N_\rho (\rho \in \Bbb F_p^*)$ Frequency 0 1 $p^{tm-2}$ $p^{tm-1}-1$ $p^{tm-2}-p^{\frac {tm-3} 2}$ $\frac {p-1} 2(p^{tm-1}+p^{\frac {tm-1} 2})$ $p^{tm-2}+p^{\frac {tm-3} 2}$ $\frac {p-1} 2(p^{tm-1}-p^{\frac {tm-1} 2})$
Complete weight enumerators of Theorem 5.1 for even $tm$
 $N_0=n-\sum_{\rho \in \Bbb F_p^*}N_\rho$ $N_\rho (\rho \in \Bbb F_p^*)$ Frequency 0 1 $p^{tm-2}$ $p^{tm-1}+(-1)^{(\frac {m(p-1)} 4+1)t}(p-1)p^{\frac {tm-2} 2}-1$ $p^{tm-2}+(-1)^{(\frac {m(p-1)} 4+1)t}p^{\frac {tm-2} 2}$ $(p-1)\big(p^{tm-1}-(-1)^{(\frac {m(p-1)} 4+1)t}p^{\frac {tm-2} 2}\big)$
 $N_0=n-\sum_{\rho \in \Bbb F_p^*}N_\rho$ $N_\rho (\rho \in \Bbb F_p^*)$ Frequency 0 1 $p^{tm-2}$ $p^{tm-1}+(-1)^{(\frac {m(p-1)} 4+1)t}(p-1)p^{\frac {tm-2} 2}-1$ $p^{tm-2}+(-1)^{(\frac {m(p-1)} 4+1)t}p^{\frac {tm-2} 2}$ $(p-1)\big(p^{tm-1}-(-1)^{(\frac {m(p-1)} 4+1)t}p^{\frac {tm-2} 2}\big)$
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