Abstract
Let m, d, e, k be fixed positive integers such that
Abstract
Let s be a fixed maximum-length binary sequence of length 2
Abstract
be a system of circular decimations of s whose decimation factors are respectively
Abstract
or respectively 2
Abstract
Then s 1 , · · · , s k−1 are maximum-length binary sequences of length 2 m − 1. Let C be the F 2 -vector space generated by all circular shifts of s, s 1 , · · · , s k−1 . Then C has an F 2 m -vector space structure, and is of dimension k over F 2 m . When k = 2, C is the Gold code. So we regard C as a Gold type code of relative dimension k. The DC component distribution of C is explicitly calculated out in the present paper.

Then s 1 , · · · , s k−1 are maximum-length binary sequences of length 2 m − 1. Let C be the F 2 -vector space generated by all circular shifts of s, s 1 , · · · , s k−1 . Then C has an F 2 m -vector space structure, and is of dimension k over F 2 m . When k = 2, C is the Gold code. So we regard C as a Gold type code of relative dimension k. The DC component distribution of C is explicitly calculated out in the present paper.

The sequence (A 0 , A 1 , · · · , A n ) is called the weight distribution of C. Given a linear code C, it is challenging to determine its weight distribution. The weight distribution of Gold codes was determined by Gold [G66-G68] . The weight distribution of Kasami codes was determined by Kasami [K66] . The weight enumerators of Gold type and Kasami type codes of higher relative dimension were determined by Berlekamp [Ber] and Kasami [K71] . The weight distribution of the p-ary analogue of Gold codes was determined by Trachtenberg [Tr] . The weight distribution of the circular decimation of the p-ary analogue of Gold codes with decimation factor 2 was determined by FengLuo [FL] . The weight distribution of the p-ary analogue of Gold type codes of relative dimension 3 was determined by Zhou-Ding-Luo-Zhang [ZDLZ] . The weight distribution of the circular decimation with decimation factor 2 of the p-ary analogue of Gold type codes of relative dimension 3 was determined by Zheng-Wang-Hu-Zeng [ZWHZ] . The weight distribution of (the p-ary analogue of) Kasami type codes of maximum relative dimension was determined by Li-Hu-Feng-Ge [LHFG] . The weight distribution of the pary analogue of Gold type codes of higher relative dimension was determined by Schmidt [Sch] . The weight distribution of some other classes of cyclic codes was determined in the papers [AL] , [BEW] , [BMC] , [BMC10] , [BMY] , [De] , [DLMZ] , [DY] , [FE] , [FM] , [KL] , [LF] , [LHFG] , [LN] , [LYL] , [LTW] , [MCE] , [MCG] , [MO] , [MR] , [MY] , [MZLF] , [RP] , [SC] , [VE] , [WTQYX] , [XI] , [XI12] , [YCD] , [YXDL] and [ZHJYC] .

Let m, d, e, k be fixed positive integers such that

Let s be a fixed maximum-length binary sequence of length 2 m −1. Let (s 1 , s 2 , · · · , s k−1 ) be a system of circular decimations of s whose decimation factors are respectively

or respectively 2

Then s 1 , · · · , s k−1 are maximum-length binary sequences of length 2 m − 1. Let C be the F 2 -vector space generated by all circular shifts of s, s 1 , · · · , s k−1 . If d = e = 1, then C is the code studied by Berlekamp [Ber] and Kasami [K71] . Let {Q a } be the system

or the system

or the system

with π being a primitive element of F 2 m . The correspondence a → c a defines an F 2 m -vector space structure on C, and C is of dimension k over F 2 m . When k = 2, C is the Gold code. So we call C a Gold type code of relative dimension k.

One can prove the following.

where

The present paper is concerned with the frequencies

The main result of the present paper is the following.

Theorem 1.2 For each j = 0, 1, · · · , k − 2, and for each ε = ±1, we have

is a Gaussian binomial coefficient.

From the above theorem one can deduce the following.

2 ).

If d = e = 1, then the weight enumerator of C is determined by Berlekamp [Ber] and Kasami [K71] . However, some extra calculations are needed to explicitly write out the coefficients of the weight enumerators in [Ber, K71] .
ENTERING BILINEAR FORMS I
In this section we shall prove Theorem 1.1.
ENTERING BILINEAR FORMS I
It is well-known that
ENTERING BILINEAR FORMS I
Let
ENTERING BILINEAR FORMS I
Then {B a } is either the system
ENTERING BILINEAR FORMS I
or the system
ENTERING BILINEAR FORMS I
or the system
ENTERING BILINEAR FORMS I
It is well-known that
ENTERING BILINEAR FORMS I
We now prove Theorem 1.1. By (2), (3) and (7), it suffices to prove the following.
ENTERING BILINEAR FORMS I
where
ENTERING BILINEAR FORMS I
Without loss of generality, we assume that {B a } is the system (4). Then
ENTERING BILINEAR FORMS I
Note that
ENTERING BILINEAR FORMS I
The theorem is proved.
ENTERING BILINEAR EQUATIONS II
In this section we shall reduce Theorem 1.2 to the following. 
ENTERING BILINEAR EQUATIONS II
It suffices to prove the following. (−1)
ENTERING BILINEAR EQUATIONS II
Similarly,
ENTERING BILINEAR EQUATIONS II
The theorem is proved.
ASSOCIATION SCHEME THEORETIC APPROACH
In this section we shall use the following theorem of Delarte-Goethals to prove Theorem 3.1.
ASSOCIATION SCHEME THEORETIC APPROACH
Theorem 4.1 ( [DG] ) Let M be an odd number, X the space of alternating bilinear forms on an M -dimension vector space over F q , Y a subspace of X, and
ASSOCIATION SCHEME THEORETIC APPROACH
Moreover, if the equality holds, then, for i
ASSOCIATION SCHEME THEORETIC APPROACH
We now the above theorem to prove Theorem 3.1.
ASSOCIATION SCHEME THEORETIC APPROACH
Let X be the space of alternating F 2 e -bilinear forms on F 2 m . Fix a system {B a }. Set
ASSOCIATION SCHEME THEORETIC APPROACH
By Theorem 2.1,
ASSOCIATION SCHEME THEORETIC APPROACH
By Delsarte-Goethals' theorem, Theorem 3.1 is proved.
NUMBER THEORETIC APPROACH
The theorem of Delarte-Goethals we used in the last section is proved by developing the theory of association schemes. To make the present paper self-contained, we shall develop a number theoretic approach, which is similar to the approach of Berlekamp [Ber] and Kasami [K71] .
NUMBER THEORETIC APPROACH
Let V s,u be the set of solutions (x 1 , x 2 , · · · , x 2u ) ∈ F 2u 2 m of one of the systems
NUMBER THEORETIC APPROACH
and
NUMBER THEORETIC APPROACH
In this section we shall use the following theorem to prove Theorem 3.1.
NUMBER THEORETIC APPROACH
We now prove Theorem 3.1. We shall make repeated use of the following q-binomial formula
NUMBER THEORETIC APPROACH
By the orthogonality of characters and Theorem 5.1, we have
NUMBER THEORETIC APPROACH
where |V 0,0 | = 1. Applying the identity 
NUMBER THEORETIC APPROACH
That is,
NUMBER THEORETIC APPROACH
Consider the equation
NUMBER THEORETIC APPROACH
Multiplying on the left by the row vector ((−1)
NUMBER THEORETIC APPROACH
, and applying the q-binomial formula, we arrive at
NUMBER THEORETIC APPROACH
Applying the q-binomial formula once more, we arrive at
NUMBER THEORETIC APPROACH
Replacing k − 1 with an arbitrary positive integer u, we arrive at
NUMBER THEORETIC APPROACH
That is,
NUMBER THEORETIC APPROACH
Now fix 0 ≤ u ≤ k − 1, and consider the equation
NUMBER THEORETIC APPROACH
Multiplying on the left by the row vector ((−1)
NUMBER THEORETIC APPROACH
, and applying the q-binomial formula as well as (12), we arrive at
NUMBER THEORETIC APPROACH
0≤h≤u−1
NUMBER THEORETIC APPROACH
Dividing both sides by 0≤h≤u−1 (4 eu − 4 eh ), we arrive at Applying the q-binomial Möbius inversion formula
NUMBER THEORETIC APPROACH
we arrive at Theorem 3.1 is proved.
SYSTEMS OF BILINEAR EQUATIONS
In this section we shall prove Theorem 5.1. We begin with the following.
SYSTEMS OF BILINEAR EQUATIONS
Theorem 6.1 The systems (9), (10), and (11) are respectively equivalent to the systems
SYSTEMS OF BILINEAR EQUATIONS
and
SYSTEMS OF BILINEAR EQUATIONS
Proof. We deal with the system (9) first. Adding 2 d -th power of the (j −1)-th equation to the j-th equation, we arrive at
SYSTEMS OF BILINEAR EQUATIONS
Adding the (j − 1)-th equation to the j-th equation in the above system, we arrive at the system (13).
SYSTEMS OF BILINEAR EQUATIONS
We now deal with the system (10). Adding 2 2d -th power of the (j − 1)-th equation to the j-th equation, we arrive at
SYSTEMS OF BILINEAR EQUATIONS
Adding the (j − 1)-th equation to the j-th equation in the above system, we arrive at the system (14).
SYSTEMS OF BILINEAR EQUATIONS
Finally we deal with the system (11). Inserting 2 m+e 2e -th power of the first equation to the system, we arrive at the system
SYSTEMS OF BILINEAR EQUATIONS
Adding the 2 −d -th power of the (j − 1)-th equation to the j-th equation in the above system, we arrive at
SYSTEMS OF BILINEAR EQUATIONS
Adding the (j − 1)-th equation to the j-th equation in the above system, we arrive at the system (15). Theorem 6.1 is proved.
SYSTEMS OF BILINEAR EQUATIONS
We now prove Theorem 5.1. If u = 1, then V s,u = V u,u trivially. Now assume that u ≥ 2. Suppose that (x 1 , x 2 , · · · , x 2u ) belongs to V u,u . We are going to show that (x 1 , x 2 , · · · , x 2u ) belongs to V s,u . By induction, we may assume that x 2u = 0. Then we may further assume that x 2u = 1. By Theorem 6.1, (x 1 ,x 2 , · · · ,x 2u ) ∈ V u−1,u . Asx 2u = 0, we see that (x 1 ,x 2 , · · · ,x 2u−2 ) ∈ V u−1,u−1 . By induction, (x 1 ,x 2 , · · · ,x 2u−2 ) ∈ V s−1,u−1 . Asx 2u = 0, we see that (x 1 ,x 2 , · · · ,x 2u ) ∈ V s−1,u . By Theorem 6.1, (x 1 , x 2 , · · · , x 2u ) belongs to V s,u . Theorem 5.1 is proved.
THE NUMBER OF BALANCED SEQUENCES
In this section we prove Theorem 1.3. We have #{c ∈ C | DC(c) = −1} 2 ). Theorem 1.3 is proved.

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