Weight distribution of cyclic codes with arbitrary number of generalized Niho type zeroes

Abstract

Cyclic codes are an important class of linear codes, whose weight distribution have been extensively studied. Most previous results obtained so far were for cyclic codes with no more than three zeroes. Inspired by the works of Li et al. (Sci China Math 53:3279–3286, 2010; IEEE Trans Inf Theory 60:3903–3912, 2014), we study two families of cyclic codes over ${\mathbb F}_p$ with arbitrary number of zeroes of generalized Niho type, more precisely ${\mathcal {C}_{(d_0,d_1,\ldots ,d_t)}^{(1)}}$ (for $p=2$) of $t+1$ zeroes, and ${\mathcal {C}_{(\widetilde{d}_1,\ldots ,\widetilde{d}_t)}^{(2)}}$ (for any prime $p$) of $t$ zeroes for any $t$. We find that the first family has at most $(2t+1)$ non-zero weights, and the second has at most $2t$ non-zero weights. Their weight distribution are also determined in the paper.

Appendices

Appendix 1: Calculation of $N_r$ for $p=2$

Let us consider $N_r$ for $r \ge 2$. We remark that $N_2,N_3$ were obtained in [18], however the computation was somewhat complicated. Here we use a different idea which enables us to compute $N_r$ in general.

1.1 Case ${\mathcal {C}_{(d_0,d_1,\ldots ,d_t)}^{(1)}}$

For this case, $N_r$ equals the number of solutions $(x_1,\ldots ,x_r) \in \left( {\mathbb F}_{q^2}^*\right) ^r$ to the equations given by (13). We write each $x_i \in {\mathbb F}_{q^2}^*$ as

$$\begin{aligned} x_i=y_i^{\bar{\triangle }}z_i, \quad y_i \in {\mathbb F}_q^*, z_i \in U, \end{aligned}$$

where $U=\{z \in {\mathbb F}_{q^2}:z \bar{z}=z^{q+1}=1\}$ is a cyclic group of order $q+1$ and $\bar{\triangle } \triangle \equiv 1 \pmod {q-1}$. Note that this representation of $x_i$ for $y_i \in {\mathbb F}_q^*, z_i \in U$ is unique. We obtain

$$\begin{aligned} x_i^{d_j}=y_i^{\bar{\triangle }d_j}z_i^{d_j}=y_iz_i^{-2jh}, \, 0 \le j \le t. \end{aligned}$$

Now denote $u_i=z_i^{-2h}$. Then $u_i \in W:=U^e$, where $e=\gcd (2h,q+1)$. It is clear that $W$ is a cyclic group of order $\frac{q+1}{e}$ and for each $u_i \in W$, there are exactly $e$ many $z_i \in U$ that satisfy the relation $u_i=z_i^{-2h}$. Using these $y_i,u_i$’s, we can write (13) as as a matrix equation

$$\begin{aligned} \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 1&{}1&{} \cdots &{}1 \\ u_1 &{} u_2 &{} \cdots &{}u_r\\ u_1^2 &{} u_2^2 &{} \cdots &{}u_r^2\\ \vdots &{} \vdots &{} \ddots &{}\vdots \\ u_1^t &{}u_2^t &{} \cdots &{}u_r^t \end{array} \right] \cdot \left[ \begin{array}{c} y_1\\ y_2\\ \vdots \\ y_r\end{array}\right] =\underline{0},\end{aligned}$$

(18)

where we solve for variables $u_i,y_i$ such that $u_i \in W$ and $y_i \in {\mathbb F}_q^*, 1 \le i \le r$.

Next we take the $q$-th power on both sides of each equation in (18) (except the first one). Noting that $y_i^q=y_i$ and $u_i^{q}=u_i^{-1}$, thus we obtain

$$\begin{aligned} \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} u_1^{-1} &{} u_2^{-1} &{} \cdots &{}u_r^{-1}\\ u_1^{-2} &{} u_2^{-2} &{} \cdots &{}u_r^{-2}\\ \vdots &{} \vdots &{} \ddots &{}\vdots \\ u_1^{-t} &{}u_2^{-t} &{} \cdots &{}u_r^{-t} \end{array} \right] \cdot \left[ \begin{array}{c} y_1\\ y_2\\ \vdots \\ y_r\end{array}\right] =\underline{0}. \end{aligned}$$

(19)

We combine the matrices in (18) and (19) together, observing that the exponent of $u_i$ in each column goes consecutively from $-t$ to $t$, this matrix behaves like a Vandermonde matrix whose rank is easy to understand. In particular if $r \le 2t+1$, then the rank of the matrix equals the number of distinct elements in the set $\{u_1,u_2, \ldots ,u_r\}$.

1.2 Partition and type

To compute the number of solutions $u_i \in W,y_i \in {\mathbb F}_q^*, 1 \le i \le r$ that satisfy both (18) and (19), we divide the solution set $u_i \in W, y_i \in {\mathbb F}_q^*$ according to how the elements $u_1,\ldots , u_r$ may match with each other. This matching will be indicated by a partition of the set $\{1,\ldots ,r \}$ into a disjoint union of subsets

$$\begin{aligned} \{1,\ldots ,r\}=\bigcup _{\mu =1}^f I_{\mu }, \end{aligned}$$

which corresponds to valid solutions $u_i,y_i$ such that $u_i=u_j$ if and only if $i,j$ belong to the same set $I_{\mu }$ for some $\mu$.

We will compute the number of solutions $y_i,u_i$ for each such partition. To illustrate the point, let us take an example.

Example 7

The partition $\{1,2, \ldots ,7\}=\{1,2\} \cup \{3,4\} \cup \{5,6,7\}$ corresponds to the subset of solutions $u_i,y_i$ of (18) and (19) such that $u_1=u_2=\tau _1,u_3=u_4=\tau _2,u_5=u_6=u_7=\tau _3$, where $\tau _1,\tau _2,\tau _3 \in W$ are distinct, and $y_1,y_2,,\ldots ,y_7 \in {\mathbb F}_q^*$. Combining this and (18), (19) we have

$$\begin{aligned} \left[ \begin{array}{c@{\quad }c@{\quad }c} 1&{} \cdots &{}1 \\ \tau _1 &{} \tau _2 &{}\tau _3\\ \tau _1^2 &{} \tau _2^2 &{} \tau _3^2 \\ \ddots &{}\vdots &{}\vdots \\ \tau _1^t &{} \tau _2^t &{} \tau _3^t \\ \tau _1^{-1} &{} \tau _2^{-1} &{}\tau _3^{-1}\\ \tau _1^{-2} &{} \tau _2^{-2} &{}\tau _3^{-2}\\ \vdots &{} \ddots &{}\vdots \\ \tau _1^{-t} &{} \tau _2^{-t} &{}\tau _3^{-t} \end{array} \right] \cdot \left[ \begin{array}{l} y_1+y_2\\ y_3+y_4\\ y_5+y_6+y_7 \end{array}\right] =\underline{0}.\end{aligned}$$

(20)

The matrix on the left has rank 3, hence we have

$$\begin{aligned} y_1+y_2=0, \quad y_3+y_4=0, \quad y_5+y_6+y_7=0. \end{aligned}$$

The number of solutions for $y_i \in {\mathbb F}_q^*, 1 \le i \le 7$ that satisfy the above equations is given by $(q-1)^2 (q-1)(q-2)$. The number of ways to choose $u_i \in W, 1 \le i \le 7$ is given by $3! \left( {\begin{array}{c}(q+1)/e\\ 3\end{array}}\right)$. Finally for each $u_i$, $1 \le i \le 7$ there are $e$ ways to choose $z_i \in U$ such that $z_i^{-2h}=u_i$. So the total number of solutions $y_i,z_i$ corresponding to this partition is given by

$$\begin{aligned} 3!(q-1)^2 (q-1)(q-2) \left( {\begin{array}{c}(q+1)/e\\ 3\end{array}}\right) e^{7}. \end{aligned}$$

$\square$

Now we resume our computation. For a given partition $\bigcup _{\mu =1}^f I_{\mu }$ of $\{1,\ldots ,r \}$, its “flag” is defined to be a vector of non-negative integers $\underline{\uplambda }=(\uplambda _1,\uplambda _2,\ldots ,)$ where $\uplambda _j=\#\{\mu : \#I_{\mu }=j\}$. The previous example has flag $(0,2,1)$, that is $\uplambda _1=0,\uplambda _2=2,\uplambda _3=1$. We make the following observations:

• (1) $\sum _{j} j \uplambda _{j}=r$;

• (2) The number of solutions $u_i,y_i$ corresponding to a partition only depends on the flag of the partition;

• (3) The total number of different partitions of $\{1,\ldots ,r\}$ for a given flag $\underline{\uplambda }$ is (by convention $0!=1$)

  $$\begin{aligned} \frac{r!}{\prod _{j}(\uplambda _{j})! (j !)^{\uplambda _{j}}}. \end{aligned}$$

1.3 Counting solutions for a partition

Let $\bigcup _{\mu =1}^f I_{\mu }=\{1,\ldots ,r\}$ be a valid partition with flag $\underline{\uplambda }$. Similar to the argument in the previous example, since $r \le 2t+1$, the matrix has full rank, we obtain the identity

$$\begin{aligned} \sum _{j \in I_{\mu }} y_{j}=0, \quad \forall \, \mu . \end{aligned}$$

(21)

Note that if some $I_{\mu }$ contains only one element $j$, this would force $y_j=0$, which is impossible since we require $y_j \ne 0$. So we assume that $\#I_{\mu } \ge 2$ for each $\mu$, or in other words $\uplambda _1=0$.

Denote by $B_{\tau }$ the number of solutions $y_1,\ldots ,y_{\tau } \in {\mathbb F}_q^*$ such that

$$\begin{aligned} y_1+\cdots +y_{\tau }=0. \end{aligned}$$

By the inclusion-exclusion principle, it is easy to obtain the formula

$$\begin{aligned} B_{\tau }=q^{-1}(q-1)^{\tau }+(-1)^{\tau }(1-q^{-1}), \quad \forall \tau \ge 2. \end{aligned}$$

The number of $y_i$’s that satisfies (21) is obviously $\prod _{j \ge 2}(B_j)^{\uplambda _j}$. Now we treat $u_i$.

The number of distinct elements in $\{u_1,\ldots ,u_r\}$ is $D:=\sum _{j} \uplambda _j$, and the number of ways to choose such $u_i$’s for this partition is given by $D! \left( {\begin{array}{c}(q+1)/e\\ D\end{array}}\right)$. On the other hand, each such $u_i$ (there are $r$ of them) gives rise to $e$ many $z_i$’s. In summary, we find that for flag $\underline{\uplambda }$, the number of solutions $y_i,z_i$ is given by

$$\begin{aligned} \frac{r!}{\prod _{j}(\uplambda _j)!(j!)^{\uplambda _j}} \left( {\begin{array}{c}\frac{q+1}{e}\\ D\end{array}}\right) D! e^{r} \prod _{j} (B_j)^{\uplambda _j}. \end{aligned}$$

Summing over all $\uplambda _j$ such that $\sum _{j \ge 2} j \uplambda _j=r$ gives the value $N_r$. This completes the proof of the formula $N_r$ for ${\mathcal {C}_{(d_0,d_1,\ldots ,d_t)}^{(1)}}$. $\square$

1.4 Case ${\mathcal {C}_{(\widetilde{d}_1,\ldots ,\widetilde{d}_t)}^{(2)}}$

Here $N_r$ is the number of solutions $(x_1,\ldots ,x_r) \in \left( {\mathbb F}_{q^2}^*\right) ^r$ to the equations given by (17). Using the same notation as before, we write each $x_i$ uniquely as

$$\begin{aligned}x_i=y_i^{\bar{\triangle }}z_i, \quad y_i \in {\mathbb F}_q^*, z_i \in U, \end{aligned}$$

and we obtain

$$\begin{aligned}x_i^{\widetilde{d}_j}=y_i^{\bar{\triangle }\widetilde{d}_j}z_i^{\widetilde{d}_j}=y_iz_i^{-(2j-1)h}=y_iu_i^{2j-1}, \quad 1 \le j \le t, \end{aligned}$$

where $u_i=z_i^{-h}$. Then $u_i \in W:=U^e$, where $e=\gcd (h,q+1)$. $W$ is a cyclic group of order $\frac{q+1}{e}$ and for each $u_i \in W$, there are exactly $e$ many $z_i \in U$ that satisfy the relation $u_i=z_i^{-h}$. Using $y_i,u_i$’s, we can write (17) as a matrix equation

$$\begin{aligned} \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} u_1 &{} u_2 &{} \cdots &{}u_r\\ u_1^3 &{} u_2^3 &{} \cdots &{}u_r^3\\ \vdots &{} \vdots &{} \ddots &{}\vdots \\ u_1^{2t-1} &{}u_2^{2t-1} &{} \cdots &{}u_r^{2t-1} \end{array} \right] \cdot \left[ \begin{array}{c} y_1\\ y_2\\ \vdots \\ y_r\end{array}\right] =\underline{0}. \end{aligned}$$

(22)

Again taking the $q$-th power on both sides of each equation we obtain

$$\begin{aligned} \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} u_1^{-1} &{} u_2^{-1} &{} \cdots &{}u_r^{-1}\\ u_1^{-3} &{} u_2^{-3} &{} \cdots &{}u_r^{-3}\\ \vdots &{} \vdots &{} \ddots &{}\vdots \\ u_1^{-2t+1} &{}u_2^{-2t+1} &{} \cdots &{}u_r^{-2t+1} \end{array} \right] \cdot \left[ \begin{array}{c} y_1\\ y_2\\ \vdots \\ y_r\end{array}\right] =\underline{0}. \end{aligned}$$

(23)

Combining the matrices in (22) and (23) together and noting that the exponent of $u_i$ in each column goes consecutively from $-2t+1$ to $2t-1$ with gap 2, we see that this matrix also behaves like a Vandermonde matrix whose rank is easy to understand. In particular if $r \le 2t$, then the rank of the matrix equals the number of distinct elements in the set $\{u_1,u_2, \ldots ,u_r\}$. This is the only property which was used in the argument for the previous case ${\mathcal {C}_{(d_0,d_1,\ldots ,d_t)}^{(1)}}$. Hence we conclude that $N_r$ could be computed in exactly the same way as before, and it is given by the formula (6) for $r \le 2t$. This completes the proof for the case ${\mathcal {C}_{(\widetilde{d}_1,\ldots ,\widetilde{d}_t)}^{(2)}}$. $\square$

Appendix 2: Calculation of $N_r$ for $p$ odd

Now $p$ is an odd prime, $q=p^m$ and $N_r$ is the number of solutions $(x_1,\ldots ,x_r) \in \left( {\mathbb F}_{q^2}^*\right) ^r$ to the equations given by (17). Let $\gamma$ be a generator of ${\mathbb F}_{q^2}^*$. Using the same notation as before, we may write each $x_i \in {\mathbb F}_{q^2}^*$ as

$$\begin{aligned} x_i=y_i^{\bar{\triangle }}z_i \epsilon _i, \quad y_i \in {\mathbb F}_q^*, z_i \in U, \epsilon _i \in \{\gamma ,1\}. \end{aligned}$$

(24)

Since

$$\begin{aligned} {\mathbb F}_q^* \bigcap U=\{1,-1\}, \end{aligned}$$

as $y_i,z_i,\epsilon _i$ run over the sets ${\mathbb F}_q^*,U$ and $\{\gamma ,1\}$ once respectively, the $x_i$ will run over ${\mathbb F}_{q^2}^*$ exactly twice. So $N_r=2^{-r}M_r$ where $M_r$ is the number of $y_i \in {\mathbb F}_q^*,z_i \in U,\epsilon _i \in \{\gamma ,1\}, 1 \le i \le r$ such that the $x_i$’s from (24) satisfy the equations (17) simultaneously. We obtain

$$\begin{aligned} x_i^{\widetilde{d}_j}=y_i^{\bar{\triangle }\widetilde{d}_j}z_i^{\widetilde{d}_j}\epsilon _i^{\widetilde{d}_j}=y_iz_i^{-(2j-1)h}\epsilon _i^{\widetilde{d}_j}=y_i\left( z_i\epsilon _i^{-(q-1)/2}\right) ^{-(2j-1)h}\epsilon _i^{\triangle (q+1)/2}, \quad 1 \le j \le t. \end{aligned}$$

Moreover, since $\epsilon _i^{\widetilde{s}_j(q-1)} \in U$,

$$\begin{aligned} x_i^{q\widetilde{d}_j}=y_iz_i^{(2j-1)h}\epsilon _i^{q\widetilde{d}_j}=y_i\left( z_i\epsilon _i^{-(q-1)/2}\right) ^{(2j-1)h}\epsilon _i^{\triangle (q+1)/2}, \quad 1 \le j \le t. \end{aligned}$$

Define $u_i=\left( z_i\epsilon _i^{-(q-1)/2}\right) ^{-h} , \xi _i=\epsilon _i^{\triangle (q+1)/2}$. Then $u_i\epsilon _i^{(q-1)h/2} \in W:=U^e$, where $e=\gcd (h,q+1)$. $W$ is a cyclic group of order $\frac{q+1}{e}$ and for each $u_i\epsilon _i^{(q-1)h/2} \in W$, there are exactly $e$ many $z_i \in U$ that satisfy the relation $u_i=\left( z_i\epsilon _i^{-(q-1)/2}\right) ^{-h}$. Using $y_i,u_i,\xi _i$’s, we can write (17) as a matrix equation

$$\begin{aligned} \left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r} u_1\xi _1 &{} u_2 \xi _2 &{} \cdots &{}u_r \xi _r\\ u_1^3 \xi _1 &{} u_2^3 \xi _2&{} \cdots &{}u_r^3 \xi _r\\ \vdots &{} \vdots &{} \ddots &{}\vdots \\ u_1^{2t-1} \xi _1 &{}u_2^{2t-1} \xi _2 &{} \cdots &{}u_r^{2t-1} \xi _r \end{array} \right] \cdot \left[ \begin{array}{c} y_1\\ y_2\\ \vdots \\ y_r\end{array}\right] =\underline{0}. \end{aligned}$$

(25)

From the $q$-th power of each equation we obtain

$$\begin{aligned} \left[ \begin{array}{r@{\quad }r@{\quad }r@{\quad }r} u_1^{-1} \xi _1&{} u_2^{-1} \xi _2&{} \cdots &{}u_r^{-1} \xi _r\\ u_1^{-3} \xi _1&{} u_2^{-3} \xi _2&{} \cdots &{}u_r^{-3} \xi _r\\ \vdots &{} \vdots &{} \ddots &{}\vdots \\ u_1^{-2t+1} \xi _1 &{}u_2^{-2t+1} \xi _2&{} \cdots &{}u_r^{-2t+1} \xi _r \end{array} \right] \cdot \left[ \begin{array}{c} y_1\\ y_2\\ \vdots \\ y_r\end{array}\right] =\underline{0}.\end{aligned}$$

(26)

Combining the matrices in (25) and (26) together and noting that each column is a multiple of $\xi _i$ and the exponent of $u_i$ goes consecutively from $-2t+1$ to $2t-1$ with gap 2, hence this matrix also behaves like a Vandermonde matrix whose rank is easy to understand. In particular if $r \le 2t$, then the rank of the matrix equals the number of distinct elements in the set $\{u_1,u_2, \ldots ,u_r\}$. Using this property, we find that for each fixed $\underline{\xi }=(\xi _1, \ldots ,\xi _r)$, the number of solutions $u_i,y_i$ that satisfy (25) and (26) is also given by the formula (6). On the other hand, each $\xi \in \{\gamma ^{\triangle (q+1)/2},1\}$ can take two distinct values, hence $\underline{\xi }$ takes $2^r$ distinct values. Taking into account that $N_r=2^{-r}M_r$, we find that this $N_r$ is exactly the same as given by the formula (6). This completes the proof for the case ${\mathcal {C}_{(\widetilde{d}_1,\ldots ,\widetilde{d}_t)}^{(2)}}$ when $p$ is odd. $\square$