On the weight distribution of second order Reed–Muller codes and their relatives

Abstract

The weight distribution of second order q-ary Reed–Muller codes have been determined by Sloane and Berlekamp (IEEE Trans. Inform. Theory, vol. IT-16, 1970) for $q=2$ and by McEliece (JPL Space Progr Summ 3:28–33, 1969) for general prime power q. Unfortunately, there were some mistakes in the computation of the latter one. This paper aims to provide a precise account for the weight distribution of second order q-ary Reed–Muller codes. In addition, the weight distributions of second order q-ary homogeneous Reed–Muller codes and second order q-ary projective Reed–Muller codes are also determined.

Appendix

In this appendix, we show that the correspondence in Table 1 holds true. Note that when q is even, the terminologies in Definition 2.3 and [5, Proposition 3.8] simply have different names and are essentially the same. Hence, it suffices to consider the q odd case. The following is a preparatory lemma.

Lemma 4.1

Let q be an odd prime power and $\lambda$ a nonsquare of ${\mathbb {F}}_q$. Then the quadratic form $x_1^2+x_2^2$ is equivalent to $\lambda (y_1^2+y_2^2)$.

Proof

It is easy to see that there exist $\lambda _1, \lambda _2 \in {\mathbb {F}}_q$, such that $\lambda _1 \ne \lambda _2$ and $\lambda _1^2+\lambda _2^2=\lambda$. The conclusion follows by applying an invertible linear transformation satisfying $x_1=\lambda _1y_1+\lambda _2y_2$ and $x_2=-\lambda _2y_1+\lambda _1y_2$. $\square$

When q is odd, we use $\eta$ to denote the quadratic multiplicative character of ${\mathbb {F}}_q$. We have the following lemma.

Lemma 4.2

Let q be an odd prime power. For $\mathbf{x}=(x_1,x_2,\ldots ,x_m)$, let $Q(\mathbf{x})=\sum _{i=1}^r a_ix_i^2$ be a quadratic form of rank r from ${\mathbb {F}}_q^m$ to ${\mathbb {F}}_q$, where each $a_i$ is a nonzero element of ${\mathbb {F}}_q$ and $\eta (\prod _{i=1}^r a_i)=\delta$. Then Q is a quadratic form of rank r and type $\tau$, where

$$\begin{aligned} \tau ={\left\{ \begin{array}{ll} -\delta &{} \text{ if } q \equiv 3 \bmod 4 \text{ and } r \equiv 2,3 \bmod 4, \\ \delta &{} \text{ otherwise. } \end{array}\right. } \end{aligned}$$

Proof

We only prove the case where the rank r is odd, since the proof of r even case is analogous. Applying Lemma 4.1, we can see that Q is equivalent to one of the following:

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathop \sum \nolimits _{i=1}^r y_i^2 &{} \text{ if } \delta =1, \\ \mathop \sum \nolimits _{i=1}^{r-1} y_i^2+\lambda y_r^2 &{} \text{ if } \delta =-1, \end{array}\right. } \end{aligned}$$

(4.1)

where $\lambda$ is a nonsquare of ${\mathbb {F}}_q$. Next, we divide the proof in two cases, in which $q \equiv 1 \bmod 4$ and $q \equiv 3 \bmod 4$, respectively.

Case I: $q \equiv 1 \bmod 4$. In this case, $-1$ is a square of ${\mathbb {F}}_q$. When $\delta =1$, by (4.1), Q is equivalent to

$$\begin{aligned} \sum _{i=1}^{\frac{r-1}{2}} z_i^2-\sum _{i=\frac{r+1}{2}}^{r-1} z_i^2+z_r^2. \end{aligned}$$

Let $w_i=z_i+z_{i+\frac{r-1}{2}}$, $w_{i+\frac{r-1}{2}}=z_i-z_{i+\frac{r-1}{2}}$ for $1 \le i \le \frac{r-1}{2}$ and $w_r=z_r$. We have that Q is equivalent to

$$\begin{aligned} \sum _{i=1}^{\frac{r-1}{2}} w_iw_{i+\frac{r-1}{2}}+w_r^2. \end{aligned}$$

Thus, Q has rank r and type 1. When $\delta =-1$, a similar argument shows that Q has rank r and type $-1$.

Case II: $q \equiv 3 \bmod 4$. In this case, $-1$ is a nonsquare of ${\mathbb {F}}_q$. We consider two subcases where $r \equiv 1 \bmod 4$ and $r \equiv 3 \bmod 4$, respectively.

When $r \equiv 1 \bmod 4$, if $\delta =1$, then by (4.1) and Lemma 4.1, Q is equivalent to

$$\begin{aligned} \sum _{i=1}^{\frac{r-1}{2}} z_i^2-\sum _{i=\frac{r+1}{2}}^{r-1} z_i^2+z_r^2. \end{aligned}$$

An analogous approach as in Case I shows that Q has rank r and type 1. Similarly, if $\delta =-1$, we can show that Q has rank r and type $-1$.

When $r \equiv 3 \bmod 4$, the situation is more involved. If $\delta =1$, then by (4.1) and Lemma 4.1, Q is equivalent to

$$\begin{aligned} \sum _{i=1}^{\frac{r-3}{2}} z_i^2-\sum _{i=\frac{r-1}{2}}^{r-3} z_i^2+z_{r-2}^2-z_{r-1}^2-z_r^2. \end{aligned}$$

Let $w_i=z_i+z_{i+\frac{r-3}{2}}$, $w_{i+\frac{r-3}{2}}=z_i-z_{i+\frac{r-3}{2}}$ for $1 \le i \le \frac{r-3}{2}$ and $w_{r-2}=z_{r-2}+z_{r-1}$, $w_{r-1}=z_{r-2}-z_{r-1}$, $w_r=z_r$. Then we can see that Q is equivalent to

$$\begin{aligned} \sum _{i=1}^{\frac{r-3}{2}} w_iw_{i+\frac{r-3}{2}}+w_{r-2}w_{r-1}+(-1)w_r^2. \end{aligned}$$

Thus, Q has rank r and type $-1$. If $\delta =-1$, then by (4.1) and Lemma 4.1, Q is equivalent to

$$\begin{aligned} \sum _{i=1}^{\frac{r-3}{2}} z_i^2-\sum _{i=\frac{r-1}{2}}^{r-3} z_i^2+z_{r-2}^2+z_{r-1}^2-z_r^2. \end{aligned}$$

Let $w_i=z_i+z_{i+\frac{r-3}{2}}$, $w_{i+\frac{r-3}{2}}=z_i-z_{i+\frac{r-3}{2}}$ for $1 \le i \le \frac{r-3}{2}$ and $w_{r-2}=z_{r-1}+z_{r}$, $w_{r-1}=z_{r-1}-z_{r}$, $w_r=z_{r-2}$. Then we can see that Q is equivalent to

$$\begin{aligned} \sum _{i=1}^{\frac{r-3}{2}} w_iw_{i+\frac{r-3}{2}}+w_{r-2}w_{r-1}+w_r^2. \end{aligned}$$

Thus, Q has rank r and type 1. Consequently, we complete the proof. $\square$

Employing Lemma 4.2 and comparing Definition 2.3 with [5, Proposition 3.8], we confirm the correspondence in Table 1.