The Weight Distributions of a Class of Cyclic Codes with Three Nonzeros over 𝔽₃

Abstract

Cyclic codes have efficient encoding and decoding algorithms. The decoding error probability and the undetected error probability are usually bounded by or given from the weight distributions of the codes. Most researches are about the determination of the weight distributions of cyclic codes with few nonzeros, by using quadratic forms and exponential sums but limited to low moments. In this paper, we focus on the application of higher moments of the exponential sums to determine the weight distributions of a class of ternary cyclic codes with three nonzeros, combining with not only quadratic forms but also MacWilliams’ identities. Another application of this paper is to emphasize the computer algebra system Magma for the investigation of the higher moments. In the end, the result is verified by one example using Matlab.

1. Introduction

Cyclic codes have a lot of applications in communication system, storage system, and computers, and they have been studied for a long time [1, 2]. The decoding error probability and the undetected error probability are closely related to the weight distributions, for example, permutation decoding, majority decoding, locator decoding, decoding from the covering polynomials, and so on [3–6]. In general the weight distributions are complicated [7] and difficult to be determined. In fact, as shown in [8, 9], the problem of computing weight distribution of a cyclic code is connected with the evaluation of certain exponential sums, which are generally hard to be determined explicitly. For more researches, refer to [10–13] for the irreducible case, [14–17] for the reducible case, and [18–21] for recent studies. For related problems in the binary case with two nonzeros especially, refer to [22–24].

In this paper, we focus on the application of higher moments of the exponential sums to determine the weight distributions of a class of ternary cyclic codes with three nonzeros, combining with not only quadratic forms but also MacWilliams’ identities, with the help of the computer algebra system Magma.

Let 𝑝 be a prime. A linear [𝑛,𝑘,𝑑;𝑝] code is a 𝑘-dimensional subspace of 𝔽^𝑛_𝑝 with minimum (Hamming) distance 𝑑. An [𝑛,𝑘] linear code 𝒞 over 𝔽_𝑝 is called cyclic if (𝑐₀,𝑐₁,…,𝑐_𝑛−1)∈𝒞 implies that (𝑐_𝑛−1,𝑐₀,𝑐₁,…,𝑐_𝑛−2)∈𝒞 where gcd(𝑛,𝑝)=1. By identifying the vector (𝑎₀,𝑎₁,…,𝑎_𝑛−1)∈𝔽^𝑛_𝑝 with 𝑎₀+𝑎₁𝑥+⋯+𝑎_𝑛−1𝑥^𝑛−1∈𝔽_𝑝[𝑥](𝑥^𝑛),−1 (1) any linear code 𝒞 of length 𝑛 over 𝔽_𝑝 represents a subset of 𝔽_𝑝[𝑥]/(𝑥^𝑛−1) which is a principle ideal domain. The fact that the code is cyclic is equivalent to the fact that the subset is an ideal. The unique monic polynomial 𝑔(𝑥) of minimum degree in this subset is the generating polynomial of 𝒞, and it is a factor of 𝑥^𝑛−1. When the ideal does not contain any smaller nonzero ideal, the corresponding cyclic code 𝒞 is called a minimal or an irreducible code. For any 𝑣=(𝑐₀,𝑐₁,…,𝑐_𝑛−1)∈𝒞, the weight of 𝑣 is 𝑤𝑡(𝑣)=#{𝑐_𝑖≠0,𝑖=0,1,…,𝑛−1}.

The weight enumerator of a code 𝒞 is defined by 1+𝐴₁𝑥+𝐴₂𝑥²+⋯+𝐴_𝑛𝑥^𝑛, (2) where 𝐴_𝑖 denotes the number of codewords with Hamming weight 𝑖. The sequence 1,𝐴₁,…,𝐴_𝑛 is called the weight distribution of the code, which is an important parameter of a linear block code.

Assume that 𝑝=3 and 𝑞=𝑝^𝑚 for an even integer 𝑚. Let 𝜋 be a primitive element of 𝔽_𝑞. In this paper, Section 2 presents the basic notations and preliminaries about cyclic codes. Section 3 determines the weight distributions of a class of cyclic codes over 𝔽₃ with nonzeros 𝜋⁻², 𝜋⁻⁴, and 𝜋⁻¹⁰, and they are verified by using Matlab. Note that the length of the cyclic code is 𝑙=𝑞−1=3^𝑚−1. Final conclusion is in Section 4. This paper is the counterpart of our other result in [19].

2. Preliminaries

In this section, relevant knowledge from finite fields [25] is presented for our study of cyclic codes. It is about the calculations of exponential sums, the sizes of cyclotomic cosets, and the ranks of certain quadratic forms. First, some known properties about the codeword weight are listed.

Let 𝑝 be an odd prime and let 𝑚 be a positive integer and 𝜋 is a primitive element of 𝔽_𝑞. Assume that the cyclic code 𝒞 over 𝔽_𝑝 has length 𝑙=𝑞−1=𝑝^𝑚−1 and nonconjugate nonzeros 𝜋^−𝑠𝜆, where 1≤𝑠_𝜆≤𝑞−2  (1≤𝜆≤𝜄). Then the codewords in 𝒞 can be expressed by 𝑐𝛼₁,…,𝛼_𝜄=𝑐₀,𝑐₁,…,𝑐_𝑙−1𝛼₁,…,𝛼_𝜄∈𝔽_𝑞, (3) where 𝑐_𝑖=∑^𝜄_𝜆=1Tr(𝛼_𝜆𝜋^𝑖𝑠𝜆)  (0≤𝑖≤𝑙−1) and Tr∶𝔽_𝑞→𝔽_𝑝 is the trace mapping from 𝔽_𝑞 to 𝔽_𝑝. Therefore the Hamming weight of the codeword 𝑐=𝑐(𝛼₁,…,𝛼_𝜄) is 𝑤_𝐻(𝑐)=#𝑖∣0≤𝑖≤𝑙−1,𝑐_𝑖𝑙≠0=𝑙−𝑝−1𝑝^𝑝−1_{𝑎=1𝑥∈𝔽∗𝑞}𝜁_𝑝^{Tr(𝑎𝑓(𝑥))}=𝑝^𝑚−1−1𝑝−1𝑝^𝑝−1_𝑎=1𝑆𝑎𝛼₁,…,𝑎𝛼_𝜄=𝑝^𝑚−1−1𝑝−1𝑝𝑅𝛼₁,…,𝛼_𝜄, (4) where 𝜁_𝑝=𝑒^{2𝜋𝑖/𝑝} (𝑖 is imaginary unit), 𝑓(𝑥)=𝛼₁𝑥^𝑠1+𝛼₂𝑥^𝑠2+⋯+𝛼_𝜄𝑥^𝑠𝜄∈𝔽_𝑞[𝑥],  𝔽^∗_𝑞=𝔽_𝑞⧵{0}, 𝑆𝛼₁,…,𝛼_𝜄=_{𝑥∈𝔽𝑞}𝜁^{Tr(𝛼1𝑥𝑠1+⋯+𝛼𝜄𝑥𝑠𝜄)}_𝑝, (5) and 𝑅(𝛼₁,…,𝛼_𝜄)=∑^𝑝−1_𝑎=1𝑆(𝑎𝛼₁,…,𝑎𝛼_𝜄).

For general functions of the form 𝑓_{𝛼,…,𝛾}(𝑥)=𝛼𝑥^𝑝𝑖+1+⋯+𝛾𝑥^𝑝𝑗+1 where 0≤𝑖,…,𝑗≤⌊𝑚/2⌋, there are quadratic forms 𝐹_{𝛼,…,𝛾}(𝑋) and corresponding symmetric matrices 𝐻_{𝛼,…,𝛾} satisfying that 𝐹_{𝛼,…,𝛾}(𝑋)=𝑋𝐻_{𝛼,…,𝛾}𝑋^𝑇=Tr(𝑓_{𝛼,…,𝛾}(𝑥)). It is known that there exists 𝑀_{𝛼,…,𝛾}∈GL_𝑚(𝔽_𝑝) such that 𝐻^′_{𝛼,…,𝛾}=𝑀_{𝛼,…,𝛾}𝐻_{𝛼,…,𝛾}𝑀^𝑇_{𝛼,…,𝛾}=diag𝑎₁,…,𝑎_{𝑟𝛼,…,𝛾},0,…,0, (6) where 𝑎_𝑖∈𝔽^∗_𝑝  (1≤𝑖≤𝑟_{𝛼,…,𝛾}) and 𝑟_{𝛼,…,𝛾}=rank𝐻_{𝛼,…,𝛾}. Let Δ=𝑎₁⋯𝑎_{𝑟𝛼,…,𝛾} (set Δ=1 for 𝑟_{𝛼,…,𝛾}=0).

Lemma 1 (Lemma 1, [15]). (i) For the quadratic form 𝐹(𝑋)=𝑋𝐻𝑋^𝑇, _{𝑋∈𝔽𝑚𝑝}𝜁_𝑝^𝐹(𝑋)=Δ𝑝𝑝^{𝑚−𝑟/2}(),𝑖𝑖𝑓𝑝≡1mod4_𝑟Δ𝑝𝑝^{𝑚−𝑟/2}𝑖𝑓𝑝≡3(mod4). (7) (ii) For 𝐴=(𝑎₁,…,𝑎_𝑚)∈𝔽^𝑚_𝑝, if  2𝑌𝐻+𝐴=0 has solution 𝑌=𝐵∈𝔽^𝑚_𝑝, then _{𝑋∈𝔽𝑚𝑝}𝜁^{𝐹(𝑋)+𝐴𝑋𝑇}_𝑝=𝜁^𝑐_{𝑝𝑋∈𝔽𝑚𝑝}𝜁_𝑝^𝐹(𝑋)1𝑤ℎ𝑒𝑟𝑒𝑐=2𝐴𝐵^𝑇∈𝔽_𝑝. (8) Otherwise ∑_𝑋∈𝔽^𝑚_𝑝𝜁^{𝐹(𝑋)+𝐴𝑋𝑇}_𝑝=0.
Here (Δ/𝑝) denotes the Legendre symbol.

References [26, Lemma 9] and [19, Lemma 3] are about the calculations of exponential sums that will be needed in the sequel. For more details, refer to [15].

The cyclotomic coset containing 𝑠 is defined to be 𝒟_𝑠=𝑠,𝑠𝑝,𝑠𝑝²,…,𝑠𝑝^𝑚𝑠−1, (9) where 𝑚_𝑠 is the smallest positive integer such that 𝑝^𝑚𝑠⋅𝑠≡𝑠(mod𝑝^𝑚−1). Reference [26, Lemma 10] is about the size of cyclotomic coset; also refer to [27] for the binary case.

For 𝑓^′_𝑑(𝑥)=𝛼₀𝑥²+𝛼₁𝑥^𝑝+1+⋯+𝛼_𝑑𝑥^𝑝𝑑+1 with corresponding quadratic form 𝐹^′_𝑑(𝑋)=Tr(𝑓^′_𝑑(𝑥))=𝑋𝐻^′_𝑑𝑋^𝑇, where (𝛼₀,𝛼₁,…,𝛼_𝑑)∈𝔽_𝑞^𝑑+1⧵{(0,0,…,0)}. Reference [19, Corollary 1] is about its rank for a special case.

Note that in Section 3, a nonzero solution of an equation system means that all the variable values are nonzero.

3. Main Result

In this section, the main result of this paper is obtained; that is, the weight distribution of the cyclic code 𝐶 with nonzeros 𝜋⁻²,𝜋^−(𝑝+1) and 𝜋^−(𝑝2+1) for the case 𝑚 is even; here 𝑝=3. For this, the first five moments of exponential sum 𝑆(𝛼,𝛽,𝛾) are computed in Sections 3.1, 3.2, and 3.3, and the MacWlliams’ identities are calculated in Section 3.4.

3.1. The First Three Moments of 𝑆(𝛼,𝛽,𝛾)

For an odd prime 𝑝 and even integer 𝑚, this subsection calculates the first three moments of the exponential sum 𝑆(𝛼,𝛽,𝛾) (5); see Lemma 2 and its other form Lemma 3.

Lemma 2 (see [19]). Let 𝑝 be an odd prime satisfying 𝑝≡3mod4 and 𝑞=𝑝^𝑚. Then there are the following results about the exponential sum 𝑆(𝛼,𝛽,𝛾) (5) corresponding to 𝑓^′₂(𝑥)=𝛼𝑥²+𝛽𝑥^𝑝+1+𝛾𝑥^𝑝2+1:(i)∑_{𝛼,𝛽,𝛾∈𝔽𝑞}𝑆(𝛼,𝛽,𝛾)=𝑝^3𝑚,(ii)∑_{𝛼,𝛽,𝛾∈𝔽𝑞}𝑆(𝛼,𝛽,𝛾)²=𝑝^3𝑚,(iii)∑_{𝛼,𝛽,𝛾∈𝔽𝑞}𝑆(𝛼,𝛽,𝛾)³=((𝑝+1)(𝑝^𝑚−1)+1)𝑝^3𝑚.

Corresponding to Lemma 1 and [19, Corollary 1], we introduce the following notations for convenience. Let 𝑁_𝜀,𝑗=𝛼,𝛽,𝛾∈𝔽³_𝑞⧵{(0,0,0)}∣𝑆𝛼,𝛽,𝛾=𝜀𝑝^{(𝑚+𝑗)/2}, (10) where 𝜀=±1 and 𝑗=0,2,4. Denote 𝑛_𝜀,𝑗=|𝑁_𝜀,𝑗|. Let 𝑁_𝜀,𝑗=𝛼,𝛽,𝛾∈𝔽³_𝑞⧵{(0,0,0)}∣𝑆𝛼,𝛽,𝛾=𝜀𝑖𝑝^{(𝑚+𝑗)/2} (11) for 𝑗=1,3, where 𝑖 is the imaginary unit. By [19, Lemma 3], set 𝑛_𝑗=𝑛_𝜀,𝑗=|𝑁_𝜀,𝑗| for 𝑗=1,3, since 𝑚−𝑗 is odd. Using those, Lemma 2 can be restated in Lemma 3 when 𝑚 is even.

Lemma 3. Let 𝑝 be an odd prime satisfying 𝑝≡3mod4, and 𝑞=𝑝^𝑚, where 𝑚 is an even integer. Then the notations defined in (10) and (11) satisfy 2𝑛₁+𝑛₃+𝑛_−1,0+𝑛_1,0+𝑛_−1,2+𝑛_1,2+𝑛_1,4+𝑛_−1,4=𝑝^3𝑚𝑛−1,_1,0−𝑛_−1,0+𝑝𝑛_1,2−𝑛_−1,2+𝑝²𝑛_1,4−𝑛_−1,4=𝑝^𝑚/2𝑝^2𝑚−1,−2𝑝𝑛₁+𝑝³𝑛₃+𝑛_1,0+𝑛_−1,0+𝑝²𝑛_1,2+𝑛_−1,2+𝑝⁴𝑛_1,4+𝑛_−1,4=𝑝^𝑚𝑝^𝑚,𝑛−1_1,0−𝑛_−1,0+𝑝³𝑛_1,2−𝑛_−1,2+𝑝⁶𝑛_1,4−𝑛_−1,4=𝑝+1𝑝^3𝑚/2𝑝^𝑚−1. (12)

Proof. Substituting the symbols of (10) and (11) to Lemma 22𝑛₁+𝑛₃+𝑛_−1,0+𝑛_1,0+𝑛_−1,2+𝑛_1,2+𝑛_1,4+𝑛_−1,4=𝑝^3𝑚−1,_{𝛼,𝛽,𝛾∈𝔽𝑞}𝑆𝛼,𝛽,𝛾=𝑝^𝑚/2𝑛_1,0−𝑛_−1,0+𝑖𝑝^(𝑚+1)/2𝑛_1,1−𝑛_−1,1+𝑝^(𝑚+2)/2𝑛_1,2−𝑛_−1,2+𝑖𝑝^(𝑚+3)/2𝑛_1,3−𝑛_−1,3+𝑝^(𝑚+4)/2𝑛_1,4−𝑛_−1,4+𝑝^𝑚=𝑝^𝑚/2𝑛_1,0−𝑛_−1,0+𝑝^(𝑚+2)/2𝑛_1,2−𝑛_−1,2+𝑝^(𝑚+4)/2𝑛_1,4−𝑛_−1,4+𝑝^𝑚=𝑝^3𝑚,_{𝛼,𝛽,𝛾∈𝔽𝑞}𝑆𝛼,𝛽,𝛾²=𝑝^𝑚𝑛_1,0+𝑛_−1,0−𝑝^𝑚+1𝑛_1,1+𝑛_−1,1+𝑝^𝑚+2𝑛_1,2+𝑛_−1,2−𝑝^𝑚+3𝑛_1,3+𝑛_−1,3+𝑝^𝑚+4𝑛_1,4+𝑛_−1,4+𝑝^2𝑚=𝑝^3𝑚,_{𝛼,𝛽,𝛾∈𝔽𝑞}𝑆𝛼,𝛽,𝛾³=𝑝^3𝑚/2𝑛_1,0−𝑛_−1,0−𝑖𝑝^3(𝑚+1)/2𝑛_1,1−𝑛_−1,1+𝑝^3(𝑚+2)/2𝑛_1,2−𝑛_−1,2−𝑖𝑝^3(𝑚+3)/2𝑛_1,3−𝑛_−1,3+𝑝^3(𝑚+4)/2𝑛_1,4−𝑛_−1,4+𝑝^3𝑚=𝑝^3𝑚/2𝑛_1,0−𝑛_−1,0+𝑝^3(𝑚+2)/2𝑛_1,2−𝑛_−1,2+𝑝^3(𝑚+4)/2𝑛_1,4−𝑛_−1,4+𝑝^3𝑚=𝑝+1𝑝^𝑚−1+1𝑝^3𝑚, (13) where the first one comes from the fact that there are 𝑝^3𝑚−1 elements in the set 𝔽³_𝑞⧵{(0,0,0)}. Also, note that 𝑆(𝛼,𝛽,𝛾)=𝑝^𝑚 when 𝛼=𝛽=𝛾=0.
Using 𝑛_𝑗=𝑛_𝜀,𝑗=|𝑁_𝜀,𝑗| for 𝑗=1,3, the result is obtained by simplification.

3.2. The Fourth Moment of 𝑆(𝛼,𝛽,𝛾)

For the fourth moment of 𝑆(𝛼,𝛽,𝛾) in the particular case of 𝑝=3, there is the following result about the number of solutions of the equation system: 𝑥²+𝑦²+𝑧²𝑥+1=0,_𝑝+1+𝑦^𝑝+1+𝑧^𝑝+1𝑥+1=0,_𝑝2+1+𝑦^𝑝2+1+𝑧^𝑝2+1+1=0 (14) in Lemma 4, which is denoted by 𝑇₄.

Lemma 4 (see [19]). Let 𝑝=3 and 𝑞=𝑝^𝑚. Then 𝑇₄=42𝑝^𝑚−3. (15)

Using Lemma 4 and 𝑀₃ in Lemma 2, 𝑀₄ is calculated in Lemma 5, where 𝑀₄=𝑀₃+(𝑞−1)𝑇₄.

Lemma 5. Let 𝑝=3 and 𝑞=𝑝^𝑚. The number of solutions of the following equation system 𝑥²+𝑦²+𝑧²+𝑤²𝑥=0,_𝑝+1+𝑦^𝑝+1+𝑧^𝑝+1+𝑤^𝑝+1𝑥=0,_𝑝2+1+𝑦^𝑝2+1+𝑧^𝑝2+1+𝑤^𝑝2+1=0 (16) is 𝑀₄=8(𝑝^𝑚−1)²+1.

Corresponding to Lemma 2, the result of the fourth moment is provided in Lemma 6 by applying Lemma 5.

Lemma 6. Let 𝑝=3 and 𝑞=𝑝^𝑚. Then _{𝛼,𝛽,𝛾∈𝔽𝑞}𝑆𝛼,𝛽,𝛾⁴=𝑀₄⋅𝑝^3𝑚=8𝑝^𝑚−1²+1𝑝^3𝑚. (17)

Corresponding to Lemma 3, Lemma 6 can be rewritten as the following corollary using the symbols of (10) and (11).

Corollary 7. Let 𝑝=3 and 𝑞=𝑝^𝑚, where 𝑚 is an even integer. Then 𝑛_1,0+𝑛_−1,0+2𝑛₁𝑝²+𝑝⁴𝑛_1,2+𝑛_−1,2+2𝑛₃𝑝⁶+𝑝⁸𝑛_1,4+𝑛_−1,4=8𝑝^𝑚−1²−𝑝^𝑚+1𝑝^𝑚. (18)

3.3. The Fifth Moment of 𝑆(𝛼,𝛽,𝛾)

For the fifth moment of 𝑆(𝛼,𝛽,𝛾), we need Magma [28] to find the number of solutions of the following equation system: 𝑥²+𝑦²+𝑧²+𝑤²+𝑢²𝑥=0,_𝑝+1+𝑦^𝑝+1+𝑧^𝑝+1+𝑤^𝑝+1+𝑢^𝑝+1𝑥=0,_𝑝2+1+𝑦^𝑝2+1+𝑧^𝑝2+1+𝑤^𝑝2+1+𝑢^𝑝2+1=0 (19) which is denoted by 𝑀₅.

The irreducible components corresponding to the projective variety defined by (19) are listed in Table 1 using Magma [28]. Every block of Table 1 contains a system of three equations (“=0” is omitted), the solutions of which satisfy (19). The union of all the solutions in each block presents the solutions of (19) exactly. Those equation systems are circulant symmetric about the variables. In general, few works are provided to deal with the moments using five variables. In this paper, Magma helps us on the reduction of such systems in Lemmas 8 and 9 and Corollary 10. For relevant knowledge of algebraic geometry, the reader is referred to [29].

Table 1 

Irreducible components corresponding to the projective variety defined by (19).

Lemma 8. Let 𝑝=3 and 𝑞=𝑝^𝑚. Then 𝑀₅=5𝑝^𝑚−18𝑝^𝑚−2𝑝−10+1. (20)

Proof. The number of nonzero solutions of 𝑥²+𝑦²=𝑥^𝑝+1+𝑦^𝑝+1=𝑥^𝑝2+1+𝑦^𝑝2+1=0 is 𝑀^′₂=𝑀₂−1=0, where 𝑀₂=1 [19], and the number of nonzero elements in 𝑀₃=#𝑥,𝑦,𝑧∈𝔽³_𝑞∣𝑥²+𝑦²+𝑧²𝑥=0,_𝑝+1+𝑦^𝑝+1+𝑧^𝑝+1𝑥=0,_𝑝2+1+𝑦^𝑝2+1+𝑧^𝑝2+1=[]=0𝑝+1𝑞−1+119 (21) is 𝑀^′₃=(𝑝+1)(𝑞−1). From Lemma 5, the number of nonzero solutions of (16) is 𝑀^′₄=𝑀₄−4𝑀^′₃−1=(𝑝^𝑚−1)(8𝑝^𝑚−4𝑝−12).
For the solutions of equation system (19), Table 1 shows that at least one of the elements 𝑥,𝑦,𝑧,𝑤,𝑢 is zero, and there are two cases to be considered.(i)If only one of the five variables is zero, the number of such solutions is 5𝑀^′₄=5(𝑝^𝑚−1)(8𝑝^𝑚−4𝑝−12).(ii)If two variables are zero, the number of such solutions is ⁵₂𝑀^′₃=10(𝑝+1)(𝑞−1). Altogether, the number of solutions of equation system (19) is 5𝑀^′₄+10𝑀^′₃+1=5𝑝^𝑚−18𝑝^𝑚−2𝑝−10+1. (22)

Applying Lemma 8, the result about the fifth moment of exponential sum 𝑆(𝛼,𝛽,𝛾) is obtained.

Lemma 9. Let 𝑝=3 and 𝑞=𝑝^𝑚. Then _{𝛼,𝛽,𝛾∈𝔽𝑞}𝑆𝛼,𝛽,𝛾⁵=𝑀₅⋅𝑝^3𝑚=5𝑝^𝑚−18𝑝^𝑚−2𝑝−10+1𝑝^3𝑚. (23)

Corollary 10. Let 𝑝=3 and 𝑞=𝑝^𝑚, where 𝑚 is an even integer. Then 𝑛_1,0−𝑛_−1,0+𝑝⁵𝑛_1,2−𝑛_−1,2+𝑝¹⁰𝑛_1,4−𝑛_−1,4=5𝑝^𝑚−18𝑝^𝑚−2𝑝−10−𝑝^2𝑚+1𝑝^𝑚/2. (24)

3.4. MacWilliams’ Identities

MacWilliams’ theorem is for the Hamming weight enumerators of linear codes over finite field [30] (we consider prime field 𝔽_𝑝). Using this theorem, Lemma 12 is provided for the weight distribution using dual code’s first few weights of Lemma 11. The two identities in Lemma 12 will combine with previous identities in final result.

Let 𝐴_𝑖 be the number of codewords of weight 𝑖 in a code 𝐶 with length 𝑙 and dimension 𝑘, where 0≤𝑖≤𝑙. Let 𝐴^′_𝑖 be the corresponding number in the dual code 𝐶^⟂. Then 𝑊_𝐶=1𝑥,𝑦𝐶^⟂𝑊_𝐶^⟂𝑥+𝑝−1𝑦,𝑥−𝑦, (25) where 𝑊_𝐶(𝑥,𝑦)=∑^𝑙_𝑖=0𝐴_𝑖𝑥^𝑙−𝑖𝑦^𝑖. Setting 𝑥=1, (25) changes to ^𝑙_𝑖=0𝐴_𝑖𝑦^𝑖=1𝑝^{𝑙𝑙−𝑘}_𝑖=0𝐴^′_𝑖1+𝑝−1𝑦^𝑙−𝑖1−𝑦^𝑖. (26) Differentiating (26) with respect to 𝑦, ^𝑙_𝑖=1𝑖𝐴_𝑖𝑦^𝑖−1=1𝑝^{𝑙𝑙−𝑘}_𝑖=0𝐴^′_𝑖(𝑙−𝑖)𝑝−11+𝑝−1𝑦^{𝑙−𝑖−1}1−𝑦^𝑖+1+𝑝−1𝑦^𝑙−𝑖𝑖(−1)1−𝑦^𝑖−1. (27) Setting 𝑦=1, the first MacWilliams’ moment identity is obtained for 𝑙≥2^𝑙_𝑖=1𝑖𝐴_𝑖𝑝^𝑘=1𝑝𝑝−1𝑙−𝐴^′₁=1𝑝𝑝−1𝑙if𝐴^′₁=0. (28) Differentiating again, ^𝑙_𝑖=1𝑖(𝑖−1)𝐴_𝑖𝑦^𝑖−2=1𝑝^{𝑙𝑙−𝑘}_𝑖=0𝐴^′_𝑖(𝑙−𝑖)(𝑙−𝑖−1)𝑝−1²×1+𝑝−1𝑦^{𝑙−𝑖−2}1−𝑦^𝑖+2(𝑙−𝑖)𝑝−11+𝑝−1𝑦^{𝑙−𝑖−1}×𝑖(−1)1−𝑦^𝑖−1+1+𝑝−1𝑦^𝑙−𝑖𝑖(𝑖−1)1−𝑦^𝑖−2. (29) Substituting 𝑦=1, the second MacWilliams’ moment identity is obtained: ^𝑙_𝑖=1𝑖(𝑖−1)𝐴_𝑖=1𝑝^𝑙−𝑘𝑙(𝑙−1)𝑝−1²𝑝^𝑙−2+2𝐴^′₂𝑝^𝑙−2if𝐴^′₁=0. (30) Differentiating for the third and fourth time, if 𝐴^′₁=𝐴^′₃=0, the fourth MacWilliams’ moment identity is ^𝑙_𝑖=1𝑖(𝑖−1)(𝑖−2)(𝑖−3)𝐴_𝑖=1𝑝^𝑙−𝑘𝑙(𝑙−1)(𝑙−2)(𝑙−3)𝑝^𝑙−4𝑝−1⁴+12𝐴^′₂(𝑙−2)(𝑙−3)𝑝−1²𝑝^𝑙−4+24𝐴^′₄𝑝^𝑙−4. (31)

Lemma 11. Let 𝑝=3, 𝑞=𝑝^𝑚. Let 𝒞 denote the cyclic code with nonzeros 𝜋⁻², 𝜋^−(𝑝+1), and 𝜋^−(𝑝2+1); the weights of the dual code 𝒞^⟂ satisfy the following: 𝐴^′₀=1,𝐴^′₁=0,𝐴^′₂=𝑝^𝑚−1,𝐴^′₃𝐴=0,_′4=𝑝^𝑚−12𝑝^𝑚−𝑝−33+𝑝^𝑚−1𝑝^𝑚−32. (32)

Proof. Below, codewords are considered in the dual code. It is easy to see that 𝐴^′₀=1 and 𝐴^′₁=0. For the codewords with weight two, if the components at the two positions have the same value, we find that 𝑀^′₂=0. Let us consider the following equation system about the positions: 𝑥²−𝑦²𝑥=0,_𝑝+1−𝑦^𝑝+1𝑥=0,_𝑝2+1−𝑦^𝑝2+1=0, (33) which should be satisfied by the coordinates of the codewords. For any 𝑦∈𝔽^∗_𝑞, 𝑥=−𝑦 is the other corresponding coordinate. That is, 𝐴^′₂=𝑝^𝑚−1.
As to weight-three codewords, there are two cases to be considered. (i)If all the values corresponding to the three coordinates of the codewords are the same, then study the solutions of the following equation system: 𝑥²+𝑦²𝑥+1=0,_𝑝+1+𝑦^𝑝+1𝑥+1=0,_𝑝2+1+𝑦^𝑝2+1+1=0. (34) From the first two equations of (34), we find that 𝑥²=𝑦²=1 contradicting the fact that 𝑥,𝑦,1 should be different.(ii)If one value is different from the other two, consider 𝑥²−𝑦²𝑥+1=0,_𝑝+1−𝑦^𝑝+1𝑥+1=0,_𝑝2+1−𝑦^𝑝2+1+1=0. (35) Solving the above system, we have 𝑥=0 contradicting the fact that the coordinates should be different from 0.Combing the above two cases, 𝐴^′₃=0.
Now, let us consider the number of codewords with weight four in three cases.
Case I. At the four positions, the components have the same value. According to the proof of Lemma 8, the number of nonzero solutions of equation system (16) is 𝑀^′₄=(𝑝^𝑚−1)(8𝑝^𝑚−4𝑝−12). For a solution (𝑥,𝑦,𝑧,𝑤) of (16), if two of them are equal, for example, 𝑧=𝑤=𝑣, then (16) becomes 𝑥²+𝑦²−𝑣²𝑥=0,_𝑝+1+𝑦^𝑝+1−𝑣^𝑝+1𝑥=0,_𝑝2+1+𝑦^𝑝2+1−𝑣^𝑝2+1=0. (36) Solving the above system, it can be found that 𝑥 or 𝑦 is zero, so the number of nonzero solutions of (36) is 0. Then all those 𝑀^′₄ nonzero solutions of (16) correspond to the codewords, where 24=4! solutions correspond to a four-tuple and each tuple corresponds to two codewords. Therefore, there are 2⋅𝑀^′₄/24=𝑀^′₄/12 codewords.
Case II. One value at the four nonzero positions is different from the other three values. Consider the solutions of the following system: 𝑥²+𝑦²+𝑧²−𝑤²𝑥=0,_𝑝+1+𝑦^𝑝+1+𝑧^𝑝+1−𝑤^𝑝+1𝑥=0,_𝑝2+1+𝑦^𝑝2+1+𝑧^𝑝2+1−𝑤^𝑝2+1=0. (37) Using Mamga [28], the irreducible components of the projective variety defined by (37) are provided by the polynomials listed in Table 2. It is easy to see that at least one of 𝑥,𝑦,𝑧,𝑤 is zero, so the solutions can not correspond to codewords.
Case III. Two values at the coordinates are the same. Let us consider 𝑥²+𝑦²−𝑧²−𝑤²𝑥=0,_𝑝+1+𝑦^𝑝+1−𝑧^𝑝+1−𝑤^𝑝+1𝑥=0,_𝑝2+1+𝑦^𝑝2+1−𝑧^𝑝2+1−𝑤^𝑝2+1=0. (38) Again the irreducible components are presented in Table 3, by which only the cases 𝑥=−𝑧, 𝑦=−𝑤, and 𝑥=−𝑤, 𝑦=−𝑧 are the possible solutions which can correspond to codewords since coordinates should be different. The number of such solutions is 2(𝑝^𝑚−1)(𝑝^𝑚−3) which corresponds to 4⋅(2(𝑝^𝑚−1)(𝑝^𝑚−3)/16)=(𝑝^𝑚−1)(𝑝^𝑚−3)/2 codewords, since every four-tuple (𝑥₀,𝑦₀,𝑧₀,𝑤₀) corresponds to 4⋅2⋅2 solutions of (38). In fact, if 𝑐 is a weight-four codeword with nonzero positions and values (𝑥₀,𝑦₀,−𝑥₀,−𝑦₀)→(1,1,−1,−1), then (𝑥₀,𝑦₀,−𝑥₀,−𝑦₀)→(−1,−1,1,1), (𝑥₀,𝑦₀,−𝑥₀,−𝑦₀)→(−1,1,1,−1), and (𝑥₀,𝑦₀,−𝑥₀,−𝑦₀)→(1,−1,−1,1) can all represent weight-four codewords.
Combing the above three cases, 𝐴^′₄=𝑀^′₄+𝑝12^𝑚−1𝑝^𝑚−32=𝑝^𝑚−12𝑝^𝑚−𝑝−33+𝑝^𝑚−1𝑝^𝑚−32. (39) The result of the lemma is obtained.

Table 2 

Irreducible components corresponding to (37).

Table 3 

Irreducible components corresponding to (38).

Lemma 12. Let 𝑝=3, 𝑞=𝑝^𝑚, where 𝑚≥6 is an even integer. The notations defined in (10) and (11) satisfy 𝑛_1,0+𝑛_−1,0+𝑝²𝑛_1,2+𝑛_−1,2+𝑝⁴𝑛_1,4+𝑛_−1,4𝑛=𝑎,_1,0+𝑛_−1,0+𝑝⁴𝑛_1,2+𝑛_−1,2+𝑝⁸𝑛_1,4+𝑛_−1,4=𝑏, (40) where 𝑎=𝑝^3𝑚−2𝑝^𝑚−1𝑝^𝑚−2𝑝−1²+2𝐴^′₂−𝑝^2(𝑚−1)𝑝−1²𝑝^3𝑚−2𝑝^2𝑚+𝑝+1𝑝−1^𝑚−1𝑝^3𝑚−1×𝑝−1²𝑝^𝑚−2−1,𝑏=𝑝^3𝑚−4𝑝^𝑚−1𝑝^𝑚−2𝑝^𝑚−3𝑝^𝑚−4𝑝−1⁴+12𝐴^′₂𝑝^𝑚−3𝑝^𝑚−4𝑝−1²+24𝐴^′₄−𝑝−1⁵𝑝^3(𝑚−1)𝑎−𝑝^4(𝑚−1)𝑝−1⁴×𝑝^3𝑚−4𝑝^2𝑚−16𝑝^𝑚+19+6𝑝−1³𝑝^2(𝑚−1)𝑎+𝑝^3(𝑚−1)𝑝−1³×𝑝^3𝑚−𝑝^2𝑚+1−4𝑝^𝑚+6−11𝑝−1²𝑝^𝑚−2𝑎+𝑝^2(𝑚−1)𝑝−1²𝑝^3𝑚−2𝑝^2𝑚𝑝+1+6𝑝−1^𝑚−1𝑝^3𝑚−1×𝑝−1⁴𝑝^{2𝑚−4−1}, (41) and 𝐴^′₂,𝐴^′₄ are defined in Lemma 11.

Proof. Define the following notations for simplification: 𝑅₀₀=𝑝^𝑚−1𝑝−1,𝑅₀=𝑝−1𝑝𝑝^𝑚/2,𝑅₂=𝑝−1𝑝𝑝^(𝑚+2)/2,𝑅₄=𝑝−1𝑝𝑝^(𝑚+4)/2. (42) The use of MacWilliams’ identities in the following paragraphs implies the condition 𝑚≥6; refer to [26, Lemma 10]. By (4) and [26, Lemma 9], 𝒞 has seven possible nonzero weights: 𝐴_𝑅00=2𝑛₁+𝑛₃,𝐴_{𝑅00−𝑅0}=𝑛_1,0,𝐴_𝑅00+𝑅0=𝑛_−1,0,𝐴_{𝑅00−𝑅2}=𝑛_1,2,𝐴_𝑅00+𝑅2=𝑛_−1,2,𝐴_{𝑅00−𝑅4}=𝑛_1,4,𝐴_𝑅00+𝑅4=𝑛_−1,4. (43) With the above notations, the first four moments of codeword weights can be computed: ^𝑙_𝑖=0𝑖𝐴_𝑖=𝑅₀₀2𝑛₁+𝑛₃+𝑅₀₀−𝑅₀𝑛_1,0+𝑅₀₀+𝑅₀𝑛_−1,0+𝑅₀₀−𝑅₂𝑛_1,2+𝑅₀₀+𝑅₂𝑛_−1,2+𝑅₀₀−𝑅₄𝑛_1,4+𝑅₀₀+𝑅₄𝑛_−1,4=𝑅₀₀2𝑛₁+𝑛₃+𝑛_1,0+𝑛_−1,0+𝑛_1,2+𝑛_−1,2+𝑛_1,4+𝑛_−1,4−𝑅₀𝑛_1,0−𝑛_−1,0+𝑅₂𝑛_1,2−𝑛_−1,2+𝑅₄𝑛_1,4−𝑛_−1,4=𝑝^3𝑚−1𝑝^𝑚−1𝑝−1−𝑝−1𝑝𝑝^𝑚/2𝑝^𝑚/2𝑝^2𝑚=𝑝−1𝑝−1^𝑚−1𝑝^3𝑚−1,_𝑙𝑖=0𝑖²𝐴_𝑖=𝑅²₀₀2𝑛₁+𝑛₃+𝑅₀₀−𝑅₀²𝑛_1,0+𝑅₀₀+𝑅₀²𝑛_−1,0+𝑅₀₀−𝑅₂²𝑛_1,2+𝑅₀₀+𝑅₂²𝑛_−1,2+𝑅₀₀−𝑅₄²𝑛_1,4+𝑅₀₀+𝑅₄²𝑛_−1,4=𝑅²₀₀2𝑛₁+𝑛₃+𝑛_1,0+𝑛_−1,0+𝑛_1,2+𝑛_−1,2+𝑛_1,4+𝑛_−1,4−2𝑅₀₀𝑅₀𝑛_1,0−𝑛_−1,0+𝑅₂𝑛_1,2−𝑛_−1,2+𝑅₄𝑛_1,4−𝑛_−1,4+𝑅²₀𝑛_1,0+𝑛_−1,0+𝑅²₂𝑛_1,2+𝑛_−1,2+𝑅²₄𝑛_1,4+𝑛_−1,4=𝑝−1²𝑝^𝑚−2×𝑛_1,0+𝑛_−1,0+𝑝²𝑛_1,2+𝑛_−1,2+𝑝⁴𝑛_1,4+𝑛_−1,4+𝑝^2(𝑚−1)𝑝−1²𝑝^3𝑚−2𝑝^2𝑚+1_𝑙𝑖=0,𝑖³𝐴_𝑖=𝑅³₀₀2𝑛₁+𝑛₃+𝑅₀₀−𝑅₀³𝑛_1,0+𝑅₀₀+𝑅₀³𝑛_−1,0+𝑅₀₀−𝑅₂³𝑛_1,2+𝑅₀₀+𝑅₂³𝑛_−1,2+𝑅₀₀−𝑅₄³𝑛_1,4+𝑅₀₀+𝑅₄³𝑛_−1,4=𝑅³₀₀2𝑛₁+𝑛₃+𝑛_1,0+𝑛_−1,0+𝑛_1,2+𝑛_−1,2+𝑛_1,4+𝑛_−1,4−3𝑅²₀₀𝑅₀𝑛_1,0−𝑛_−1,0+𝑅₂𝑛_1,2−𝑛_−1,2+𝑅₄𝑛_1,4−𝑛_−1,4+3𝑅₀₀𝑅²₀𝑛_1,0+𝑛_−1,0+𝑅²₂𝑛_1,2+𝑛_−1,2+𝑅²₄𝑛_1,4+𝑛_−1,4−𝑅³₀𝑛_1,0−𝑛_−1,0+𝑅³₂𝑛_1,2−𝑛_−1,2+𝑅³₄𝑛_1,4−𝑛_−1,4=(𝑝−1)³𝑝^2(𝑚−1)×𝑛_1,0+𝑛_−1,0+𝑝²𝑛_1,2+𝑛_−1,2+𝑝⁴𝑛_1,4+𝑛_−1,4+𝑝^3(𝑚−1)𝑝−1³𝑝^3𝑚−𝑝^2𝑚+1−4𝑝^𝑚+6_𝑙𝑖=0,𝑖⁴𝐴_𝑖=𝑅⁴₀₀2𝑛₁+𝑛₃+𝑅₀₀−𝑅₀⁴𝑛_1,0+𝑅₀₀+𝑅₀⁴𝑛_−1,0+𝑅₀₀−𝑅₂⁴𝑛_1,2+𝑅₀₀+𝑅₂⁴𝑛_−1,2+𝑅₀₀−𝑅₄⁴𝑛_1,4+𝑅₀₀+𝑅₄⁴𝑛_−1,4=𝑅⁴₀₀2𝑛₁+𝑛₃+𝑛_1,0+𝑛_−1,0+𝑛_1,2+𝑛_−1,2+𝑛_1,4+𝑛_−1,4−4𝑅³₀₀𝑅₀𝑛_1,0−𝑛_−1,0+𝑅₂𝑛_1,2−𝑛_−1,2+𝑅₄𝑛_1,4−𝑛_−1,4+6𝑅²₀₀𝑅²₀𝑛_1,0+𝑛_−1,0+𝑅²₂𝑛_1,2+𝑛_−1,2+𝑅²₄𝑛_1,4+𝑛_−1,4−4𝑅₀₀𝑅³₀𝑛_1,0−𝑛_−1,0+𝑅³₂𝑛_1,2−𝑛_−1,2+𝑅³₄𝑛_1,4−𝑛_−1,4+𝑅⁴₀𝑛_1,0+𝑛_−1,0+𝑅⁴₂𝑛_1,2+𝑛_−1,2+𝑅⁴₄𝑛_1,4+𝑛_−1,4=𝑝−1⁵𝑝^3(𝑚−1)×𝑛_1,0+𝑛_−1,0+𝑝²𝑛_1,2+𝑛_−1,2+𝑝⁴𝑛_1,4+𝑛_−1,4+𝑝−1⁴𝑝^2𝑚−4×𝑛_1,0+𝑛_−1,0+𝑝⁴𝑛_1,2+𝑛_−1,2+𝑝⁸𝑛_1,4+𝑛_−1,4+𝑝^4(𝑚−1)𝑝−1⁴𝑝^3𝑚−4𝑝^2𝑚−16𝑝^𝑚+19. (44)(45)(46)(47) From (30), (44), and (45) we have ^𝑙_𝑖=1𝑖(𝑖−1)𝐴_𝑖=1𝑝^𝑙−𝑘𝑙(𝑙−1)𝑝−1²𝑝^𝑙−2+2𝐴^′₂𝑝^𝑙−2=𝑝^3𝑚−2𝑙(𝑙−1)𝑝−1²+2𝐴^′₂=^𝑙_𝑖=1𝑖²𝐴_𝑖−^𝑙_𝑖=1𝑖𝐴_𝑖=𝑝−1²𝑝^𝑚−2𝑛_1,0+𝑛_−1,0+𝑝²𝑛_1,2+𝑛_−1,2+𝑝⁴𝑛_1,4+𝑛_−1,4+𝑝^2(𝑚−1)𝑝−1²𝑝^3𝑚−2𝑝^2𝑚−𝑝+1𝑝−1^𝑚−1𝑝^3𝑚−1, (48) and the first one of (40) is obtained after simplification.
According to the MacWilliams’ fourth moment identity (31), (44), (45), (46), and (47), ^𝑙_𝑖=1𝑖(𝑖−1)(𝑖−2)(𝑖−3)𝐴_𝑖=1𝑝^𝑙−𝑘𝑙(𝑙−1)(𝑙−2)(𝑙−3)𝑝^𝑙−4𝑝−1⁴+12𝐴^′₂(𝑙−2)(𝑙−3)𝑝−1²𝑝^𝑙−4+24𝐴^′₄𝑝^𝑙−4=𝑝^3𝑚−4𝑝^𝑚−1𝑝^𝑚−2𝑝^𝑚−3𝑝^𝑚−4𝑝−1⁴+12𝐴^′₂𝑝^𝑚−3𝑝^𝑚−4𝑝−1²+24𝐴^′₄=^𝑙_𝑖=1𝑖⁴𝐴_𝑖−6^𝑙_𝑖=1𝑖³𝐴_𝑖+11^𝑙_𝑖=1𝑖²𝐴_𝑖−6^𝑙_𝑖=1𝑖𝐴_𝑖=𝑝−1⁴𝑝^2𝑚−4×𝑛_1,0+𝑛_−1,0+𝑝⁴𝑛_1,2+𝑛_−1,2+𝑝⁸𝑛_1,4+𝑛_−1,4+𝑝−1⁵𝑝^3(𝑚−1)𝑎+𝑝^4(𝑚−1)𝑝−1⁴𝑝^3𝑚−4𝑝^2𝑚−16𝑝^𝑚+19−6𝑝−1³𝑝^2(𝑚−1)𝑎+𝑝^3(𝑚−1)𝑝−1³𝑝^3𝑚−𝑝^2𝑚+1−4𝑝^𝑚+6+11𝑝−1²𝑝^𝑚−2𝑎+𝑝^2(𝑚−1)𝑝−1²𝑝^3𝑚−2𝑝^2𝑚𝑝+1−6𝑝−1^𝑚−1𝑝^3𝑚−1, (49) and the second one of (40) is obtained after simplification.

3.5. Weight Distribution of 𝒞

In this subsection, the parameters defined in (10) and (11) are calculated in Lemma 13, and the weight distribution of the cyclic code 𝒞 is determined in Theorem 15.

Lemma 13. Let 𝑝=3, 𝑞=𝑝^𝑚 where 𝑚≥6 is an even integer. The notations defined in (10) and (11) satisfy 𝑛_1,0=−𝑏+𝑐₆−𝑎𝑝²−𝑎𝑝³−𝑎𝑝⁴−𝑎𝑝⁵−𝑏𝑝²+𝑐₁𝑝⁶+𝑐₂𝑝⁶+𝑐₃𝑝³+𝑐₃𝑝⁵−𝑐₄𝑝²−𝑐₄𝑝⁴+𝑐₅𝑝²×−2𝑝⁶+2𝑝⁴+2𝑝²−2⁻¹,𝑛_−1,0=−𝑏−𝑐₆−𝑎𝑝²−𝑎𝑝³−𝑎𝑝⁴−𝑎𝑝⁵−𝑏𝑝²+𝑐₁𝑝⁶−𝑐₂𝑝⁶+𝑐₃𝑝³+𝑐₃𝑝⁵+𝑐₄𝑝²+𝑐₄𝑝⁴+𝑐₅𝑝²×−2𝑝⁶+2𝑝⁴+2𝑝²−2⁻¹,2𝑛₁=−𝑏−𝑐₅+𝑎𝑝³−𝑐₃𝑝³𝑝²−𝑝⁴,𝑛_1,2=𝑐₄−𝑐₆+𝑎𝑝−𝑐₅𝑝+𝑎𝑝²+𝑎𝑝⁴+𝑎𝑝⁵−𝑐₁𝑝⁵−𝑐₂𝑝⁴−𝑐₃𝑝²−𝑐₃𝑝⁴+𝑐₄𝑝⁴×2𝑝⁷−4𝑝⁵+2𝑝³⁻¹,𝑛_−1,2=−𝑐₄−𝑐₆−𝑎𝑝+𝑐₅𝑝−𝑎𝑝²−𝑎𝑝⁴−𝑎𝑝⁵+𝑐₁𝑝⁵−𝑐₂𝑝⁴+𝑐₃𝑝²+𝑐₃𝑝⁴+𝑐₄𝑝⁴×2𝑝⁷−4𝑝⁵+2𝑝³⁻¹,2𝑛₃=𝑏−𝑐₅+𝑎𝑝−𝑐₃𝑝𝑝⁴−𝑝⁶,𝑛_1,4=−𝑏+𝑐₄−𝑐₅−𝑐₆+𝑎𝑝−𝑐₃𝑝+𝑎𝑝²+𝑎𝑝³+𝑎𝑝⁴−𝑏𝑝²−𝑐₁𝑝⁴−𝑐₂𝑝²−𝑐₃𝑝³+𝑐₄𝑝²×2𝑝¹⁰−2𝑝⁸−2𝑝⁶+2𝑝⁴⁻¹,𝑛_−1,4=−𝑏−𝑐₄−𝑐₅+𝑐₆+𝑎𝑝−𝑐₃𝑝+𝑎𝑝²+𝑎𝑝³+𝑎𝑝⁴−𝑏𝑝²−𝑐₁𝑝⁴+𝑐₂𝑝²−𝑐₃𝑝³−𝑐₄𝑝²×2𝑝¹⁰−2𝑝⁸−2𝑝⁶+2𝑝⁴⁻¹, (50) where 𝑐₁=𝑝^3𝑚−1,𝑐₂=𝑝^𝑚/2𝑝^2𝑚,𝑐−1₃=𝑝^𝑚𝑝^𝑚−1,𝑐₄=𝑝+1𝑝^3𝑚/2𝑝^𝑚,𝑐−1₅=8𝑝^𝑚−1²−𝑝^𝑚+1𝑝^𝑚,𝑐₆=5𝑝^𝑚−18𝑝^𝑚−2𝑝−10−𝑝^2𝑚+1𝑝^𝑚/2, (51) and 𝑎,𝑏 are defined in Lemma 12.

Proof. From Lemma 3, Corollaries 7 and 10, and Lemma 12, the following equations hold: 2𝑛₁+𝑛₃+𝑛_−1,0+𝑛_1,0+𝑛_−1,2+𝑛_1,2+𝑛_1,4+𝑛_−1,4=𝑐₁,𝑛_1,0−𝑛_−1,0+𝑝𝑛_1,2−𝑛_−1,2+𝑝²𝑛_1,4−𝑛_−1,4=𝑐₂,−2𝑝𝑛₁+𝑝³𝑛₃+𝑛_1,0+𝑛_−1,0+𝑝²𝑛_1,2+𝑛_−1,2+𝑝⁴𝑛_1,4+𝑛_−1,4=𝑐₃,𝑛_1,0−𝑛_−1,0+𝑝³𝑛_1,2−𝑛_−1,2+𝑝⁶𝑛_1,4−𝑛_−1,4=𝑐₄,𝑛_1,0+𝑛_−1,0+2𝑛₁𝑝²+𝑝⁴𝑛_1,2+𝑛_−1,2+2𝑛₃𝑝⁶+𝑝⁸𝑛_1,4+𝑛_−1,4=𝑐₅,𝑛_1,0−𝑛_−1,0+𝑝⁵𝑛_1,2−𝑛_−1,2+𝑝¹⁰𝑛_1,4−𝑛_−1,4=𝑐₆,𝑛_1,0+𝑛_−1,0+𝑝²𝑛_1,2+𝑛_−1,2+𝑝⁴𝑛_1,4+𝑛_−1,4𝑛=𝑎,_1,0+𝑛_−1,0+𝑝⁴𝑛_1,2+𝑛_−1,2+𝑝⁸𝑛_1,4+𝑛_−1,4=𝑏. (52) Solving the above equation system, the result is obtained.

Lemma 14. Let 𝑝=3, 𝑞=𝑝^𝑚, where 𝑚≥6 is an even integer. The number of solutions of the equation system 𝑥²+𝑦²+𝑧²+𝑤²+𝑢²+𝑣²𝑥=0,_𝑝+1+𝑦^𝑝+1+𝑧^𝑝+1+𝑤^𝑝+1+𝑢^𝑝+1+𝑣^𝑝+1𝑥=0,_𝑝2+1+𝑦^𝑝2+1+𝑧^𝑝2+1+𝑤^𝑝2+1+𝑢^𝑝2+1+𝑣^𝑝2+1=0, (53) is 𝑛_1,0+𝑛_−1,0+𝑝⁶𝑛_1,2+𝑛_−1,2+𝑝¹²𝑛_1,4+𝑛_−1,4−𝑝³𝑛_1,1+𝑛_−1,1−𝑝⁹𝑛_1,3+𝑛_−1,3+𝑝^3𝑚. (54)

Proof. The sixth moment of exponential sum 𝑆(𝛼,𝛽,𝛾) satisfies _{𝛼,𝛽,𝛾∈𝔽𝑞}𝑆𝛼,𝛽,𝛾⁶=𝑝^3𝑚𝑛_1,0+𝑛_−1,0+𝑝^3(𝑚+2)𝑛_1,2+𝑛_−1,2+𝑝^3(𝑚+4)𝑛_1,4+𝑛_−1,4−𝑝^3(𝑚+1)𝑛_1,1+𝑛_−1,1−𝑝^3(𝑚+3)𝑛_1,3+𝑛_−1,3+𝑝^6𝑚=𝑀₆⋅𝑝^3𝑚, (55) where 𝑀₆ is the number of solutions of (53). Solving the above equation for 𝑀₆, the result is obtained.

Equation (53) considers the case for 6 variables. Using the seventh moment of 𝑆(𝛼,𝛽,𝛾), the number of solutions can be calculated when there are 7 variables and so forth.

Theorem 15. Let 𝑝=3, 𝑞=𝑝^𝑚, where 𝑚≥6 is an even integer. The cyclic code 𝒞 with nonzeros 𝜋⁻²,𝜋^−(𝑝+1) and 𝜋^−(𝑝2+1) has seven nonzero weights: 𝐴_{𝑝𝑚−1(𝑝−1)}=2𝑛₁+𝑛₃,𝐴_{𝑝𝑚−1(𝑝−1)−((𝑝−1)/𝑝)𝑝𝑚/2}=𝑛_1,0,𝐴_{𝑝𝑚−1(𝑝−1)+((𝑝−1)/𝑝)𝑝𝑚/2}=𝑛_−1,0,𝐴_{𝑝𝑚−1(𝑝−1)−((𝑝−1)/𝑝)𝑝(𝑚+2)/2}=𝑛_1,2,𝐴_{𝑝𝑚−1(𝑝−1)+((𝑝−1)/𝑝)𝑝(𝑚+2)/2}=𝑛_−1,2,𝐴_{𝑝𝑚−1(𝑝−1)−((𝑝−1)/𝑝)𝑝(𝑚+4)/2}=𝑛_1,4,𝐴_{𝑝𝑚−1(𝑝−1)+((𝑝−1)/𝑝)𝑝(𝑚+4)/2}=𝑛_−1,4, (56) where 𝜋 is a primitive element of the finite field 𝔽_𝑞.

It is interesting to consider the symmetric property about the weights of the cyclic code 𝒞. If there is a weight of the form 𝑝^𝑚−1(𝑝−1)+((𝑝−1)/𝑝)𝑝^{(𝑚+2𝑖)/2}  (𝑖=0,1,2), then there is a weight of the form 𝑝^𝑚−1(𝑝−1)−((𝑝−1)/𝑝)𝑝^{(𝑚+2𝑖)/2}. It seems that the weights are symmetric about the value 𝑝^𝑚−1(𝑝−1) which is also a weight of 𝒞. This phenomenon may be explained by the fact that, for symmetric matrices with corresponding quadratic residue Δ, there are symmetric matrices with corresponding nonresidue Δ^′ (Lemma 1). As the following example illustrates, in general the higher the value 𝑖 the less the number of corresponding weights.

Example 16. Let 𝑝=3, 𝑞=𝑝^𝑚, where 𝑚=6. The cyclic code 𝒞 has nonzeros 𝜋⁻², 𝜋^−(𝑝+1), and 𝜋^−(𝑝2+1), where 𝜋 is a primitive element of the finite field 𝔽_𝑞. Using Theorem 15, it has seven nonzero weights: 𝐴₄₈₆=124245576,𝐴₄₆₈𝐴=128432304,₅₀₄=119277522,𝐴₄₃₂𝐴=8591310,₅₄₀=6866496,𝐴₃₂₄=4732,𝐴₆₄₈=2548, (57) which are verified by using Matlab.

4. Conclusion

Since the weight distributions play an important role in the applications of cyclic codes, this paper focuses on the determination of those for a class of cyclic codes with three nonzeros. Relevant results received a lot of attention by using methods with lower moments of exponential sums. Here we try to apply higher moments to deal with the problem.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported in part by the National Basic Research Program of China under Grants 2013CB338004 and 2012CB316100 and the National Natural Science Foundation of China under Grant 61271222.