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:
[1]

L. D. Baumert and R. J. McEliece, Weights of irreducible cyclic codes, Inf. Control, 20 (1972), 158-175.  doi: 10.1016/S0019-9958(72)90354-3.  Google Scholar

[2] B. BerndtR. Evans and K. Williams, Gauss and Jacobi Sums, John Wiley & Sons company, New York, 1998.   Google Scholar
[3]

A. R. Calderbank and J. M. Goethals, Three-weight codes and association schemes, Philips J. Res., 39 (1984), 143-152.   Google Scholar

[4]

A. R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc., 18 (1986), 97-122.  doi: 10.1112/blms/18.2.97.  Google Scholar

[5]

C. CarletC. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inf. Theory, 51 (2005), 2089-2102.  doi: 10.1109/TIT.2005.847722.  Google Scholar

[6]

C. Ding, Codes from Difference Sets, World Scientific, Singapore, 2015.  Google Scholar

[7]

C. Ding, Linear codes from some 2-designs, IEEE Trans. Inf. Theory, 61 (2015), 3265-3275.  doi: 10.1109/TIT.2015.2420118.  Google Scholar

[8]

C. DingT. HellesethT. Klove and X. Wang, A general construction of authentication codes, IEEE Trans. Inf. Theory, 53 (2007), 2229-2235.  doi: 10.1109/TIT.2007.896872.  Google Scholar

[9]

C. DingC. LiN. Li and Z. Zhou, Three-weight cyclic codes and their weight distributions, Discr. Math., 339 (2016), 415-427.  doi: 10.1016/j.disc.2015.09.001.  Google Scholar

[10]

C. DingY. LiuC. Ma and L. Zeng, The weight distributions of the duals of cyclic codes with two zeros, IEEE Trans. Inf. Theory, 57 (2011), 8000-8006.  doi: 10.1109/TIT.2011.2165314.  Google Scholar

[11]

C. Ding, J. Luo and H. Niederreiter, Two-weight codes punctured from irreducible cyclic codes, in Proceedings of the First Worshop on Coding and Cryptography (eds. Y. Li, et al. ), World Scientific, Singapore, 4 (2008), 119-124. doi: 10.1142/9789812832245_0009.  Google Scholar

[12]

C. Ding and H. Niederreiter, Cyclotomic linear codes of order 3, IEEE Trans. Inf. Theory, 53 (2007), 2274-2277.  doi: 10.1109/TIT.2007.896886.  Google Scholar

[13]

C. Ding and X. Wang, A coding theory construction of new systematic authentication codes, Theor. Comp. Sci., 330 (2005), 81-99.  doi: 10.1016/j.tcs.2004.09.011.  Google Scholar

[14]

C. Ding and J. Yang, Hamming weights in irreducible cyclic codes, Discr. Math., 313 (2013), 434-446.  doi: 10.1016/j.disc.2012.11.009.  Google Scholar

[15]

C. Ding and J. Yin, Algebraic constructions of constant composition codes, IEEE Trans. Inf. Theory, 51 (2005), 1585-1589.  doi: 10.1109/TIT.2005.844087.  Google Scholar

[16]

K. Ding and C. Ding, Binary linear codes with three weights, IEEE Comm. Letters, 18 (2014), 1879-1882.   Google Scholar

[17]

K. Ding and C. Ding, A class of two-weight and three-weight codes and their applications in secret sharing, IEEE Trans. Inf. Theory, 61 (2015), 5835-5842.  doi: 10.1109/TIT.2015.2473861.  Google Scholar

[18]

C. LiQ. Yue and F. W. Fu, A construction of several classes of two-weight and three-weight linear codes, Appl. Alg. Eng. Comm. Comp., 28 (2017), 11-30.  doi: 10.1007/s00200-016-0297-4.  Google Scholar

[19]

S. LiT. Feng and G. Ge, On the weight distribution of cyclic codes with Niho exponents, IEEE Trans. Inf. Theory, 60 (2014), 3903-3912.  doi: 10.1109/TIT.2014.2318297.  Google Scholar

[20] R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley Publishing Inc., 1983.   Google Scholar
[21]

J. Luo and T. Helleseth, Constant composition codes as subcodes of cyclic codes, IEEE Trans. Inf. Theory, 57 (2011), 7482-7488.  doi: 10.1109/TIT.2011.2161631.  Google Scholar

[22]

C. MaL. ZengY. LiuD. Feng and C. Ding, The weight enumerator of a class of cyclic codes, IEEE Trans. Inf. Theory, 57 (2011), 397-402.  doi: 10.1109/TIT.2010.2090272.  Google Scholar

[23]

F. J. MacWilliamsC. L. Mallows and N. J. A. Sloane, Generalizations of Gleason's theorem on weight enumerators of self-dual codes, IEEE Trans. Inf. Theory, 18 (1972), 794-805.  doi: 10.1109/tit.1972.1054898.  Google Scholar

[24]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977.  Google Scholar

[25]

C. TangN. LiY. QiZ. Zhou and T. Helleseth, Linear codes with two or three weights from weakly regular bent functions, IEEE Trans. Inf. Theory, 62 (2016), 1166-1176.  doi: 10.1109/TIT.2016.2518678.  Google Scholar

[26]

G. Vega, The weight distribution of an extended class of reducible cyclic codes, IEEE Trans. Inf. Theory, 58 (2012), 4862-4869.  doi: 10.1109/TIT.2012.2193376.  Google Scholar

[27]

M. Xiong, The weight distributions of a class of cyclic codes, Finite Fields Appl., 18 (2012), 933-945.  doi: 10.1016/j.ffa.2012.06.001.  Google Scholar

[28]

J. YangM. XiongC. Ding and J. Luo, Weight distribution of a class of cyclic codes with arbitrary number of zeros, IEEE Trans. Inf. Theory, 59 (2013), 5985-5993.  doi: 10.1109/TIT.2013.2266731.  Google Scholar

[29]

S. Yang and Z. Yao, Complete weight enumerators of a class of linear codes, Discr. Math., 340 (2017), 729-739.  doi: 10.1016/j.disc.2016.11.029.  Google Scholar

[30]

S. YangX. Kong and C. Tang, A construction of linear codes and their complete weight enumerators, Finite Fields Appl., 48 (2017), 196-226.  doi: 10.1016/j.ffa.2017.08.001.  Google Scholar

[31]

J. Yuan and C. Ding, Secret sharing schemes from three classes of linear codes, IEEE Trans. Inf. Theory, 52 (2006), 206-212.  doi: 10.1109/TIT.2005.860412.  Google Scholar

[32]

X. ZengL. HuW. JiangQ. Yue and X. Cao, The weight distribution of a class of p-ary cyclic codes, Finite Fields Appl., 16 (2010), 56-73.  doi: 10.1016/j.ffa.2009.12.001.  Google Scholar

[33]

Z. ZhouN. LiC. Fan and T. Helleseth, Linear codes with two or three weights from quadratic Bent functions, Des. Codes Cryptogr., 81 (2016), 283-295.  doi: 10.1007/s10623-015-0144-9.  Google Scholar

show all references

References:
[1]

L. D. Baumert and R. J. McEliece, Weights of irreducible cyclic codes, Inf. Control, 20 (1972), 158-175.  doi: 10.1016/S0019-9958(72)90354-3.  Google Scholar

[2] B. BerndtR. Evans and K. Williams, Gauss and Jacobi Sums, John Wiley & Sons company, New York, 1998.   Google Scholar
[3]

A. R. Calderbank and J. M. Goethals, Three-weight codes and association schemes, Philips J. Res., 39 (1984), 143-152.   Google Scholar

[4]

A. R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc., 18 (1986), 97-122.  doi: 10.1112/blms/18.2.97.  Google Scholar

[5]

C. CarletC. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inf. Theory, 51 (2005), 2089-2102.  doi: 10.1109/TIT.2005.847722.  Google Scholar

[6]

C. Ding, Codes from Difference Sets, World Scientific, Singapore, 2015.  Google Scholar

[7]

C. Ding, Linear codes from some 2-designs, IEEE Trans. Inf. Theory, 61 (2015), 3265-3275.  doi: 10.1109/TIT.2015.2420118.  Google Scholar

[8]

C. DingT. HellesethT. Klove and X. Wang, A general construction of authentication codes, IEEE Trans. Inf. Theory, 53 (2007), 2229-2235.  doi: 10.1109/TIT.2007.896872.  Google Scholar

[9]

C. DingC. LiN. Li and Z. Zhou, Three-weight cyclic codes and their weight distributions, Discr. Math., 339 (2016), 415-427.  doi: 10.1016/j.disc.2015.09.001.  Google Scholar

[10]

C. DingY. LiuC. Ma and L. Zeng, The weight distributions of the duals of cyclic codes with two zeros, IEEE Trans. Inf. Theory, 57 (2011), 8000-8006.  doi: 10.1109/TIT.2011.2165314.  Google Scholar

[11]

C. Ding, J. Luo and H. Niederreiter, Two-weight codes punctured from irreducible cyclic codes, in Proceedings of the First Worshop on Coding and Cryptography (eds. Y. Li, et al. ), World Scientific, Singapore, 4 (2008), 119-124. doi: 10.1142/9789812832245_0009.  Google Scholar

[12]

C. Ding and H. Niederreiter, Cyclotomic linear codes of order 3, IEEE Trans. Inf. Theory, 53 (2007), 2274-2277.  doi: 10.1109/TIT.2007.896886.  Google Scholar

[13]

C. Ding and X. Wang, A coding theory construction of new systematic authentication codes, Theor. Comp. Sci., 330 (2005), 81-99.  doi: 10.1016/j.tcs.2004.09.011.  Google Scholar

[14]

C. Ding and J. Yang, Hamming weights in irreducible cyclic codes, Discr. Math., 313 (2013), 434-446.  doi: 10.1016/j.disc.2012.11.009.  Google Scholar

[15]

C. Ding and J. Yin, Algebraic constructions of constant composition codes, IEEE Trans. Inf. Theory, 51 (2005), 1585-1589.  doi: 10.1109/TIT.2005.844087.  Google Scholar

[16]

K. Ding and C. Ding, Binary linear codes with three weights, IEEE Comm. Letters, 18 (2014), 1879-1882.   Google Scholar

[17]

K. Ding and C. Ding, A class of two-weight and three-weight codes and their applications in secret sharing, IEEE Trans. Inf. Theory, 61 (2015), 5835-5842.  doi: 10.1109/TIT.2015.2473861.  Google Scholar

[18]

C. LiQ. Yue and F. W. Fu, A construction of several classes of two-weight and three-weight linear codes, Appl. Alg. Eng. Comm. Comp., 28 (2017), 11-30.  doi: 10.1007/s00200-016-0297-4.  Google Scholar

[19]

S. LiT. Feng and G. Ge, On the weight distribution of cyclic codes with Niho exponents, IEEE Trans. Inf. Theory, 60 (2014), 3903-3912.  doi: 10.1109/TIT.2014.2318297.  Google Scholar

[20] R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley Publishing Inc., 1983.   Google Scholar
[21]

J. Luo and T. Helleseth, Constant composition codes as subcodes of cyclic codes, IEEE Trans. Inf. Theory, 57 (2011), 7482-7488.  doi: 10.1109/TIT.2011.2161631.  Google Scholar

[22]

C. MaL. ZengY. LiuD. Feng and C. Ding, The weight enumerator of a class of cyclic codes, IEEE Trans. Inf. Theory, 57 (2011), 397-402.  doi: 10.1109/TIT.2010.2090272.  Google Scholar

[23]

F. J. MacWilliamsC. L. Mallows and N. J. A. Sloane, Generalizations of Gleason's theorem on weight enumerators of self-dual codes, IEEE Trans. Inf. Theory, 18 (1972), 794-805.  doi: 10.1109/tit.1972.1054898.  Google Scholar

[24]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977.  Google Scholar

[25]

C. TangN. LiY. QiZ. Zhou and T. Helleseth, Linear codes with two or three weights from weakly regular bent functions, IEEE Trans. Inf. Theory, 62 (2016), 1166-1176.  doi: 10.1109/TIT.2016.2518678.  Google Scholar

[26]

G. Vega, The weight distribution of an extended class of reducible cyclic codes, IEEE Trans. Inf. Theory, 58 (2012), 4862-4869.  doi: 10.1109/TIT.2012.2193376.  Google Scholar

[27]

M. Xiong, The weight distributions of a class of cyclic codes, Finite Fields Appl., 18 (2012), 933-945.  doi: 10.1016/j.ffa.2012.06.001.  Google Scholar

[28]

J. YangM. XiongC. Ding and J. Luo, Weight distribution of a class of cyclic codes with arbitrary number of zeros, IEEE Trans. Inf. Theory, 59 (2013), 5985-5993.  doi: 10.1109/TIT.2013.2266731.  Google Scholar

[29]

S. Yang and Z. Yao, Complete weight enumerators of a class of linear codes, Discr. Math., 340 (2017), 729-739.  doi: 10.1016/j.disc.2016.11.029.  Google Scholar

[30]

S. YangX. Kong and C. Tang, A construction of linear codes and their complete weight enumerators, Finite Fields Appl., 48 (2017), 196-226.  doi: 10.1016/j.ffa.2017.08.001.  Google Scholar

[31]

J. Yuan and C. Ding, Secret sharing schemes from three classes of linear codes, IEEE Trans. Inf. Theory, 52 (2006), 206-212.  doi: 10.1109/TIT.2005.860412.  Google Scholar

[32]

X. ZengL. HuW. JiangQ. Yue and X. Cao, The weight distribution of a class of p-ary cyclic codes, Finite Fields Appl., 16 (2010), 56-73.  doi: 10.1016/j.ffa.2009.12.001.  Google Scholar

[33]

Z. ZhouN. LiC. Fan and T. Helleseth, Linear codes with two or three weights from quadratic Bent functions, Des. Codes Cryptogr., 81 (2016), 283-295.  doi: 10.1007/s10623-015-0144-9.  Google Scholar

Table 1.  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
$(p-1)p^{tm-2}$ $p^{tm-1}-1$
$(p-1)(p^{tm-2}-p^{\frac {tm-3} 2})$ $\frac {p-1} 2(p^{tm-1}+p^{\frac {tm-1} 2})$
$(p-1)(p^{tm-2}+p^{\frac {tm-3} 2})$ $\frac {p-1} 2(p^{tm-1}-p^{\frac {tm-1} 2})$
Table 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
$(p-1)p^{tm-2}$ $p^{tm-1}+(-1)^{(\frac {m(p-1)} 4+1)t}(p-1)p^{\frac {tm-2} 2}-1$
$(p-1)\big(p^{tm-2}+(-1)^{(\frac {m(p-1)} 4+1)t}p^{\frac {tm-2} 2}\big)$ $(p-1)\big(p^{tm-1}-(-1)^{(\frac {m(p-1)} 4+1)t}p^{\frac {tm-2} 2}\big)$
Table 3.  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
$(p-1)p^{tm-2}$ $p^{tm-1}-1$
$(p-1)p^{tm-2}+(-1)^{\frac {(p-1)(tm+3)} 4}p^{\frac {tm-1} 2}+p^{\frac {tm-3} 2}$ $\frac {p-1} 2(p^{tm-1}+p^{\frac {tm-1} 2})$
$(p-1)p^{tm-2}+(-1)^{\frac {(p-1)(tm+3)} 4}p^{\frac {tm-1} 2}-p^{\frac {tm-3} 2}$ $\frac {p-1} 2(p^{tm-1}-p^{\frac {tm-1} 2})$
Table 4.  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
$(p-1)p^{tm-2}$ $\frac {p+1} 2p^{tm-1}-\frac {p-1} 2 p^{\frac {tm-2} 2}-1$
$(p-1)p^{tm-2}+2p^{\frac {tm-2} 2}$ $\frac {p-1} 2\big(p^{tm-1}+p^{\frac {tm-2} 2}\big)$
Table 5.  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
$(p-1)p^{tm-2}$ $\frac {p+1} 2p^{tm-1}+\frac {p-1} 2 p^{\frac {tm-2} 2}-1$
$(p-1)p^{tm-2}-2p^{\frac {tm-2} 2}$ $\frac {p-1} 2\big(p^{tm-1}-p^{\frac {tm-2} 2}\big)$
Table 6.  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
$p^{tm-2}$ $p^{tm-1}-1$
$p^{tm-2}-p^{\frac {tm-3} 2}$ $\frac {p-1} 2(p^{tm-1}+p^{\frac {tm-1} 2})$
$p^{tm-2}+p^{\frac {tm-3} 2}$ $\frac {p-1} 2(p^{tm-1}-p^{\frac {tm-1} 2})$
Table 7.  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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