New classes of NMDS codes with dimension 3

Abstract

The singleton defect of an [n, k, d] linear code $\mathcal{C}$ is defined as $s(\mathcal{C})=n-k+1-d$. Codes with $s({\mathcal {C}})=s({\mathcal {C}}^{\bot })=1$ are called near maximum distance separable (NMDS) codes. It is known that an $[n,3,n-3]$ NMDS code is equivalent to an (n, 3)-arc in PG(2, q). In this paper, by adding some suitable projective points into some known $(q+5,3)$-arcs in PG(2, q), we obtain two families of $[q+7,3,q+4]$ NMDS codes for even prime power q and a family of $[q+6,3,q+3]$ NMDS codes for odd prime power q. In addition, when $q=2^m$ and m is odd, by adding m suitable projective points into the maximum arcs in PG(2, q), we obtain a family of $[q+m+2,3,q+m-1]$ NMDS codes over ${\mathbb {F}}_q$, from which we further induce a family of NMDS codes with parameters $[q^t+m+2,3,q^t+m-1]$ over the extension field ${\mathbb {F}}_{q^t}$ for any odd integer t. All the resulting NMDS codes in this paper are shown to be linearly inequivalent to the NMDS codes constructed from elliptic curves, and their weight distributions are completely determined.

Appendices

Appendix A

Define ${\mathcal {V}}=\{ v \in {\mathbb {F}}_{2^m} {\setminus } \{1\}: \textrm{Tr}(v)=\textrm{Tr}(v^3)=\textrm{Tr}(v^{-3})=1 \}$ for odd m, and ${\mathcal {W}}=\{ w \in {\mathbb {F}}_{2^m}: w^3 \ne 1, \textrm{Tr}(w^3)=0, \textrm{Tr}(w)=\textrm{Tr}(w^{-1})=1 \}$ for even m. In the appendix, we want to prove that $\mathcal{V}\ne \emptyset$ and $\mathcal{W}\ne \emptyset$.

Note that $\textrm{Tr}(0)=0$ and $\textrm{Tr}(x)=0$ or 1 for any $x \in {{\mathbb {F}}}_{2^m}^*$, thus we have

$$\begin{aligned} \#{\mathcal {V}}&= \frac{1}{8}\sum \limits _{\begin{array}{c} {x \in {\mathbb {F}}_{2^m}^{*}} \\ {x \ne 1} \end{array}}\left[ \Big (1-(-1)^{\textrm{Tr}(x)}\Big ) \Big (1-(-1)^{\textrm{Tr}(x^3)}\Big ) \Big (1-(-1)^{\textrm{Tr}(x^{-3})}\Big )\right] \nonumber \\&=\frac{1}{8} \sum \limits _{x \in {\mathbb {F}}_{2^m}^{*} }\left[ \Big (1-(-1)^{\textrm{Tr}(x)}\Big ) \Big (1-(-1)^{\textrm{Tr}(x^3)}\Big ) \Big (1-(-1)^{\textrm{Tr}(x^{-3})}\Big )\right] -1\nonumber \\&=\frac{1}{8}N_v-1, \end{aligned}$$

(19)

and

$$\begin{aligned} \#{\mathcal {W}}&= \frac{1}{8} \sum \limits _{\begin{array}{c} {x \in {\mathbb {F}}_{2^m}^{*}} \\ {x^3 \ne 1} \end{array}} \left[ \Big (1+(-1)^{\textrm{Tr}(x^3)}\Big ) \Big (1-(-1)^{\textrm{Tr}(x)}\Big ) \Big (1-(-1)^{\textrm{Tr}(x^{-1})}\Big )\right] \nonumber \\&\ge \frac{1}{8} \sum \limits _{x \in {\mathbb {F}}_{2^m}^{*} } \left[ \Big (1+(-1)^{\textrm{Tr}(x^3)}\Big ) \Big (1-(-1)^{\textrm{Tr}(x)}\Big ) \Big (1-(-1)^{\textrm{Tr}(x^{-1})}\Big )\right] -3\nonumber \\&=\frac{1}{8}N_w-3, \end{aligned}$$

(20)

where $N_v$ and $N_w$ denote the value of the sum form in (19) and (20) respectively.

In order to evaluate $\#\mathcal{V}$ and $\#\mathcal{W}$, we need the following exponential sums:

$$\begin{aligned} K_{m}&=\sum _{x \in {\mathbb {F}}_{2^{m}}^{*}}(-1)^{{\text {Tr}}_{m}( x+x^{-1})}, ~~~ A_{m} =\sum _{x \in {\mathbb {F}}_{2^{m}}}(-1)^{{\text {Tr}}_{m}( x^{3}+ x)}, \\ B_{m}&=\sum _{x \in {\mathbb {F}}_{2^{m}}^{*}}(-1)^{{\text {Tr}}_{m}( x^{3}+ x^{-1})},~~~ R_{m} =\sum _{x \in {\mathbb {F}}_{2^{m}}}(-1)^{{\text {Tr}}_{m}( x^{3})}, \\ S_{m}&=\sum _{x \in {\mathbb {F}}_{2^{m}}^{*}}(-1)^{{\text {Tr}}_{m}( x^{3}+ x + x^{-1})},~~~ T_{m} =\sum _{x \in {\mathbb {F}}_{2^{m}}^{*}}(-1)^{{\text {Tr}}_{m}( x^{3}+ x + x^{-3})}. \end{aligned}$$

Some useful results about $K_m, A_m, B_m$ and $R_m$ have been obtained in [9] and [18, Section IV].

Lemma 14

([9, 18])

• (1) $|K_{m}| \le 2\sqrt{2^m}$, $|A_{m}| \le 2\sqrt{2^m}$ and $|B_{m}| \le 4\sqrt{2^m}$ for any m.

• (2) $R_{m}= {\left\{ \begin{array}{ll} 0, &{} \text{ if } ~ m ~\text{ is } \text{ odd }; \\ (-1)^{\frac{m}{2}+1}2^{\frac{m}{2}+1}, &{} \text{ if } ~m ~\text{ is } \text{ even }. \end{array}\right. }$

By the following important result about exponential sums, we can obtain the similar conclusions about $S_m$ and $T_m$.

Lemma 15

([26]) Let ${\overline{{\mathbb {F}}}}_{2^m}$ be the algebraic closure of ${\mathbb {F}}_{2^m}$, and $r(x)=f(x)/g(x)$ be a rational function in ${\mathbb {F}}_{2^m}(x)$ that satisfies the condition

$$\begin{aligned}r(x)\ne h(x)^2+h(x),~~\mathrm{for~any}~h(x)\in {\overline{{\mathbb {F}}}}_{2^m}(x).\end{aligned}$$

Let s be the number of distinct roots of g(x) in ${\overline{{\mathbb {F}}}}_{2^m}$. If $\chi (a)$ denotes a nontrivial character of ${\mathbb {F}}_{2^m}$, then we have

$$\begin{aligned} \left| \sum \limits _{x\in L}\chi (r(x))\right| \le \big (\max (\deg f,\deg g)+s^*-2\big )\sqrt{2^m}+\delta , \end{aligned}$$

(21)

where the sum $\sum$ runs over all $x\in {\mathbb {F}}_{2^m}$ excluding the zeros of g(x); $s^*=s$ and $\delta =1$ when $\deg f\le \deg g$, and $s^*=s+1$ and $\delta =0$ otherwise.

Then we can immediately obtain the following results.

Lemma 16

$|S_{m}| \le 4\sqrt{2^m}$ and $|T_{m}| \le 6\sqrt{2^m}$ for any positive integer m.

Now we are ready to demonstrate the existence of v and w in the corresponding finite fields ${\mathbb {F}}_{2^m}$.

Lemma 17

$\#\mathcal{V}>0$ for any odd integer $m\ge 5$.

Proof

Since $\gcd (\pm 3,2^m-1)=1$ for any odd m, then $x^{\pm 3}$ are permutations over ${\mathbb {F}}_{2^m}^*$, thus by Lemma 4, we have

$$\begin{aligned}\sum \limits _{x \in {\mathbb {F}}_{2^m}^*}(-1)^{\textrm{Tr}(x)}=\sum \limits _{x \in {\mathbb {F}}_{2^m}^*}(-1)^{\textrm{Tr}(x^3)}=\sum \limits _{x \in {\mathbb {F}}_{2^m}^*}(-1)^{\textrm{Tr}(x^{-3})}=-1.\end{aligned}$$

By simple computation, we obtain

$$\begin{aligned} N_v&=\sum \limits _{x \in {\mathbb {F}}_{2^m}^{*} } \Big [\left( 1-(-1)^{\textrm{Tr}(x)}\right) \left( 1-(-1)^{\textrm{Tr}(x^3)}\right) \left( 1-(-1)^{\textrm{Tr}(x^{-3})}\right) \Big ]\\&=2^m-1 - (-1)\times 3 + \sum \limits _{x \in {\mathbb {F}}_{2^m}^{*} } (-1)^{\textrm{Tr}(x^3+x)} + \sum \limits _{x \in {\mathbb {F}}_{2^m}^{*} } (-1)^{\textrm{Tr}(x+x^{-3})} \\&\quad + \sum \limits _{x \in {\mathbb {F}}_{2^m}^{*} } (-1)^{\textrm{Tr}(x^3+x^{-3})} - \sum \limits _{x \in {\mathbb {F}}_{2^m}^{*} } (-1)^{\textrm{Tr}(x^3+x+x^{-3})}\\&=2^m+1+A_m+B_m+K_m-T_m. \end{aligned}$$

By (19), Lemmas 14 and 16, we have

$$\begin{aligned} \#\mathcal{V}&=\frac{1}{8}N_v-1\\&\ge \frac{1}{8}\big (2^m+1-14\sqrt{2^m}\big )-1\\&=\frac{1}{8}\big (2^m-14\sqrt{2^m}-7\big ). \end{aligned}$$

Clearly $\#\mathcal{V}>0$ when $m\ge 9$. When $m=5$, let $\xi$ be a generator of ${\mathbb {F}}_{2^5}^*$ satisfying $\xi ^5+\xi ^2+1=0$. Then it is easy to verify $\xi ^3\in \mathcal{V}$ by Magma. When $m=7$, let $\tau$ be a generator of ${\mathbb {F}}_{2^7}^*$ satisfying $\tau ^7+\tau +1=0$. Then $\tau ^7\in \mathcal{V}$ by Magma. $\square$

Lemma 18

$\#\mathcal{W}>0$ for any even integer $m\ge 6$.

Proof

Since $\gcd (-1, 2^m-1)=1$, then by Lemma 4, we have

$$\begin{aligned}\sum \limits _{w \in {\mathbb {F}}_{2^m}}(-1)^{\textrm{Tr}(w)}=\sum \limits _{w \in {\mathbb {F}}_{2^m}}(-1)^{\textrm{Tr}(w^{-1})}=0.\end{aligned}$$

Similar to Lemma 17, by simple computation, we obtain

$$\begin{aligned} N_w&=\sum \limits _{x \in {\mathbb {F}}_{2^m}^{*} } \Big [\left( 1+(-1)^{\textrm{Tr}(x^3)}\right) \left( 1-(-1)^{\textrm{Tr}(x)}\right) \left( 1-(-1)^{\textrm{Tr}(x^{-1})}\right) \Big ]\\&=2^m-1-(-1)\times 2+\sum \limits _{x\in {\mathbb {F}}_{2^m}^{*} } \left( (-1)^{\textrm{Tr}(x^3)} - (-1)^{\textrm{Tr}(x^3+x)} - (-1)^{\textrm{Tr}(x^3+x^{-1})}\right. \\&\left. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+ (-1)^{\textrm{Tr}(x+x^{-1})} + (-1)^{\textrm{Tr}(x^3+x+x^{-1})}\right) \\&=2^m+1+R_m-A_m-B_m+K_m+S_m. \end{aligned}$$

By (20), Lemmas 14 and 16, we have

$$\begin{aligned} \#\mathcal{W}&\ge \frac{1}{8}N_w-3\\&\ge \frac{1}{8}\big (2^m+1-14\sqrt{2^m}\big )-3\\&=\frac{1}{8}\big (2^m-14\sqrt{2^m}-23\big ). \end{aligned}$$

Clearly $\#\mathcal{W}>0$ when $m\ge 8$. When $m=6$, let $\zeta$ be a generator of ${\mathbb {F}}_{2^6}^*$ satisfying $\zeta ^6+\zeta ^5+1=0$. Then it is easy to verify that $\zeta ^5\in \mathcal{W}$ by Magma. $\square$

Appendix B

Let $q=2^m$ with an odd positive integer m. In the appendix, we want to show the existence of $\lambda \in {\mathbb {F}}_q^*$ that satisfies the following two conditions:

• (1) $\lambda , \lambda ^2, \cdots , \lambda ^{2^{m-1}}$ are different from each other;

• (2) for each $s \in \left[ \frac{m-1}{2}\right] ,$

  $$\begin{aligned} \textrm{Tr}\left( \frac{\lambda ^{2^s+7}+\lambda ^{7\cdot 2^s+1}+\lambda ^{6\cdot 2^s+2}+\lambda ^{2^{s+1}+6}}{\lambda ^{12}+\lambda ^{6\cdot 2^{s+1}}}\right) =1. \end{aligned}$$

The following table is obtained by Magma, showing the total number of $\lambda$’s satisfying the above two conditions over ${\mathbb {F}}_q$. It is seen that the number of $\lambda$ increases with the increase of m (Table 2). Therefore we can conjecture that such $\lambda$ always exists when $m\ge 3$.

Table 2 The total number of $\lambda$’s over ${\mathbb {F}}_{2^m}$