# Minimum Distance of New Generalizations of the Punctured Binary Reed-Muller Codes

Motivated by applications in combinatorial design theory and constructing LCD codes, C. Ding et al DLX introduced cyclic codes (q,m,h) and (q,m,h) over F_q as new generalization and version of the punctured binary Reed-Muller codes. In this paper, we show several new results on minimum distance of (q,m,h) and (q,m,h) which are generalization or improvement of previous results given in DLX.

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## 1 Introduction

In 1954, Reed (R ) and Muller (M ) constructed independently a kind of binary linear codes (Reed-Muller codes). The punctured binary Reed-Muller codes are cyclic and have been generalized into ones over arbitrary finite fields (see DGM ; KLP and others). Such codes and their variants have applications not only in error correcting, but also in secret sharing, data storage systems (locally testable or locally decodable codes) and computational complexity theory. Recently, C. Ding et al DLX present new generalization and version of the punctured binary Reed-Muller codes motivated by their applications in combinatorial designs and constructing LCD codes (linear code with complement dual).

Let be a power of a prime number, , be a primitive element of the finite field , which means . Each nonzero element in can be expressed uniquely by with and . For any integer with , let

 a=a0+a1q+⋯+am−1qm−1,0≤ai≤q−1,

be the -adic expansion of . The Hamming -weight of

is defined by the Hamming weight of vector

. Namely,

 wtq(a)=♯{i:0≤i≤m−1,ai≠0}.
###### Definition 1.1.

Let be a power of a prime number, , , and . The cyclic codes and over are ideals of the ring with the generating polynomial

 gq,m,h(x)=n−1∏a=1wtq(a)≤h(x−αa)∈Fq[x]

and

 ¯¯¯gq,m,h(x)=(x−1)lcm(gq,m,h(x),ˆgq,m,h(x))

respectively, where is the reciprocal polynomial of .

Namely, let

 I(q,m,h)={1≤a≤n−1:wtq(a)≤h}
 I_(q,m,h)={n−a:a∈I(q,m,h)}.

Then the set of zeros of and are

 Z(q,m,h)={αa:a∈I(q,m,h)}

and

 ¯¯¯¯Z(q,m,h)=Z(q,m,h)∪Z_(q,m,h)∪{1}

respectively, where .

The length of the codes and is . Let and be the dimension of and over . Then

 k(q,m,h)=n−deg(gq,m,h(x))=n−h∑i=1(q−1)i(ni).

On the other hand, if , then , and . The dimension has also determined for (DLX , Theorem 26 and 27). In this paper, we focus on the minimum (Hamming) distance and of the codes and . The following results have been proved in DLX .

###### Theorem 1.2.

Let be a power of a prime number, , and . Then

(1)

(DLX , Theorem 3). Particularly, .

(2)

for (DLX , Theorem 25-27).

(3)

(reaches the lower bound in BEW ) and for all are distance-optimal (by sphere-packing bound) (DLX , Corollary 4).

(4)

(reaches the lower bound in BEW ) (DLX , Theorem 6).

In section 2 of this paper, we proved the following new results.

(I)

We provide a sufficient condition for a divisor of such that and (Theorem 2.1). In many cases, we can find such divisor so that the upper bound of can be improved and an upper bound of is presented.

As one of direct consequence of this general result , we have showed that

(II)

If and , then and (Theorem 2.2). Namely, and reach their lower bounds given by Theorem 1.2 (1) and (2) if . C. Ding et al DLX raised open problem 1: Is it true that ? From Theorem 1.2 we know that this is true if and any , , or . Theorem 2.2 provide new evidence in case.

In section 3 we give more specific consideration for case. By Theorem 1.2 we know that for ,

 q+1≤d(q,m,1)≤2q−1 , ¯¯¯d(q,m,1)≥2(q+1)

And by Theorem 2.2 we have and if is even. We get the following new result on upper bound of .

(III)

For , and the codes are distance-optimal by sphere-packing bound (Theorem 3.1). for all (Theorem 3.3). For , for sufficient large (Theorem 3.2).

(IV)

As an application of Theorem 2.1, we provide a simple sufficient condition for a divisor of such that and (Theorem 3.4). For , we make a table for such so that the upper bounds and are improved.

In last section, we make conclusion and raise some open problems for further research.

At the end of this section, we remark that C. Ding et al DLX also introduced the extended code of and show that the minimum distance is . Thus any result on can be shifted to the one on directly.

## 2 General Case

Let , , , and

 I(q,m,h)={a:1≤a≤n−1,1≤wtq(a)≤h}
 I_(q,m,h)={n−a:a∈I(q,m,h)}
 ¯¯¯I(q,m,h)=I(q,m,h)∪I_(q,m,h)∪{0}.

Then the set of zeros of the cyclic codes and is and respectively. For , we call and belong to a same -cyclotomic class, if there exists such that . The set is divided into -cyclotomic classes. For any , we defined , where is the least non-negative residue of modulo (). If is the -adic expansion of , , then , so that . Thus is divided into disjoint -cycloomic classes . For each , we take . The set is called a representative system of . Usually we take to be the smallest integer in . With these notations we know that

 ℧(q,m,h)={c(x)=n−1∑i=0cixi∈Fq[x]/(xn−1):c(αa)=0 for all a∈R(q,m,h)}
 ¯¯¯¯℧(q,m,h)={c(x)=n−1∑i=0cixi∈Fq[x]/(xn−1):c(1)=c(αa)=c(α−a)=0 for all a∈R(q,m,h)}

After all these preparation, we present the following general result.

###### Theorem 2.3.

Let be a power of a prime number, , , and . For a divisor of , if and the following condition (*) holds,

(*)

for all

then for any integer , and .

###### Proof.

Let be a divisor of satisfying the condition (*). Let

 M=ml , n=ef , N=qM−1=eF

where . Consider the following polynomial

 c(x)=N−1∑i=0cixi=xN−1xF−1∈Fq[x]/(xN−1).

We claim that .

Let . Then

 c(x)∈℧(q,m,h) ⇔c(α)=0 for all a∈I(q,M,h)={a:1≤a≤N−1,1≤wtq(a)≤h} ⇔αa is not a zero of xF−1 % for all a∈I(q,M,h).

But

 αa is a zero of xF−1 ⇔αaF=1 ⇔N|aF since the order of α is N ⇔e|a since N=eF.

Therefore,

 c(x)∈℧(q,m,h)⇔e∤a for all a∈I(q,m,h) (2.1)

For each , we know that , so that has the following -adic expansion

 a=M−1∑i=0aiqi, (0≤ai≤q−1).

If there exists such that , let . Then , and

 wtq(a′)≤wtq(a−aiqi)+wtq(aiqi)≤(h−1)+1=h.

After finite step of this procedure, we get an integer such that , and . Therefore, and , since . Then we get that if and only if . This implies that if the condition (*) holds, then the right-hand side of (2.1) is true, so that . From and

 c(x)=xN−1xF−1=x(e−1)F+x(e−2)F+⋯+xF+1, deg(c(x))=(e−1)F

we know that is a non-zero codeword in and the Hamming weight is . Therefore .

Next we consider

 ¯¯c(x)=c(x)(x−1)=(x−1)(xN−1)xF−1.

If the condition (*) holds we have proved that for all . Moreover, by similar argument, it can be shown that if the condition (*) holds, then for all since if and only if . Finally, . Therefore, for all which means that . From we know that is a non-zero codeword in and the Hamming weight . Therefore, . This completes the proof of Theorem 2.1.

Remark:

(1)

Since , any divisor of satisfying the condition (*) should be at least . On the other hand, if , then Theorem 2.1 presents a better upper bound of than the one in Theorem 1.2(1).

(2)

For any integer with , if and only if since . This means that if and belong to the same -cyclotomic class, then if and only if . Let be the partition of with () being -cyclotomic classes, is a representative set of this partition. Then for all is equivalent to the following condition:

(R)

for all .

Moreover, consider as a partial order set with respect to the divisibility order : if and only if . Let be the set of maximal elements of . It is easy to see that both of the condition (*) and (R) are equivalent to the following condition

(M)

for all .

###### Example 2.4.

Take , , , . Theorem 1.2 gives , . For , we have

 R(3,4,2) = {1,2}∪{a+3b:a,b∈{1,2}}∪{a+9b:1≤b≤a≤2} = {1,2,4,5,7,8,10,11,20},

, , is a divisor of , and for all . By Theorem 2.1, we get and .

For , , we have

 R(3,6,2) = R(3,4,2)∪{a+27b:1≤b≤a≤2} = {1,2,4,5,7,8,10,11,20}∪{28,29,58}

. Take , by Theorem 2.1, we get and .

Ding et al. raised several open problems in DLX . One of them is : Is it true that (the lower bound given in Theorem 1.2)? Theorem 1.2 shows that this is true for four cases: , , , and . As an application of Theorem 2.1, we give the following generalization of the first three cases which show more evidence for this open problem.

###### Theorem 2.5.

Let be a power of a prime number, , . Then for each , and .

###### Proof.

Take , in Theorem 2.1. Then , . For ,

 wtq(te)=wtq(h∑i=0tqi)=h+1>h.

Therefore, is not a divisor of any number in since . By Theorem 2.1 and 1.2 we get and . ∎

## 3 h=1 case

In this section we deal with case more precisely. From the first two sections we know that

(I)

For and , , .

(II)

For , and for , and .

(III)

For and even number , and .

Now we present an upper bound of . Firstly we consider case.

###### Theorem 3.6.

For all , and the code is distance optimal.(Remark that ).

###### Proof.

We know that . The parameters of binary code is where , . If , the sphere-packing bound gives

 2(n+1)2=21+2m=2n−k≥3∑i=0(ni)=1+n+12n(n−1)+16n(n−1)(n−2), (3.1)

which is equivalent to . Let . It can be checked that . This implies that for , and which contradicts to the inequality (3.1) given by the sphere-packing bound. Therefore and the codes for are distance-optimal.

The sphere-packing can also be used to obtain an upper bound of for .

###### Theorem 3.7.

For any fixed , there exists a constant such that

for all odd integers

.

###### Proof.

The parameters of the cyclic code is , where

 n=qm−1 , k=n−|I_(q,m,1)|=n−(1+2(q−1)m) , d=¯¯¯d(q,m,1).

Suppose that . The sphere-packing bound gives

 q1+2(q−1)m=qn−k≥2q−1∑i=0(q−1)i(ni). (3.2)

The last term of the right hand side is

 (q−1)2q−1(qm−12q−1)=M(q)q(2q−1)m+O(q)q(2q−2)m , M(q)=(q−1)2q−1(2q−1)!>0.

When is fixed and , we know that the equality (3.2) cannot hold for sufficient large since . This completes the proof of Theorem 3.2. ∎

By more careful estimation, it would be obtained an explicit value of

. The case is easy.

###### Theorem 3.8.

For any odd integer , .

###### Proof.

If , the sphere-packing bound gives

 3(n+1)4=34m+1≤5∑λ=02λ(nλ)%(n=3m−1 ) (3.3)

which, by an elementary computation, is equivalent to

 4n5−75n4−80n3−390n2−104n−30≤0. (3.4)

But when , and the left-hand side of (3.4) is

 n5(4−75n−80n2−390n3−104n4−30n5) ≥ n5(4−7526−80262−390263−104264−30265)>n5(4−3.3)>0.

Therefore, the equality (3.3) does not hold for any odd number . This completes the proof of Theorem 3.3. ∎

Next, we show that the upper bound can be improved in many cases by using Theorem 2.1 and a remarkable fact: is independent of . At the same cases we also present an upper bound of which is smaller than . From we know that , where .

###### Theorem 3.9.

Let be a power of a prime number.

(1)

Let , and . If the order of modulo is an odd integer , then and for all odd integer .

(2)

Let be an odd positive integer. If is a divisor of and , then and .

###### Proof.
(1)

By assumption and we know that the order of modulo is and then . From we know that for any . Then the conclusion is derived from Theorem 2.1 and Remark (2) after Theorem 2.1.

(2)

It is a direct consequence of (1). The order of modulo is a divisor of , therefore is odd and , .

###### Remark 3.10.
(1)

For , the order of modulo is two. We obtain the previous results: for even number , and .

(2)

The following facts in elementary number theory may be helpful to judge if the order modulo is odd for , and . We denote by the order of modulo . Namely, is the least integer such that .

(F1)

, where is the Euler function defined by

 ϕ(e)=♯{i:1≤i≤e,gcd(i,e)=1}.
(F2)

Let , where are distinct prime numbers, , . Then is odd if and only if is odd for all () since .

(F3)

For , . Then is odd if and only if , which means that . For , where is an odd prime and , . We know that if , then

 Op2(b)=Op(b) or Op(b)p , Opa(b)=Op2(b)⋅pa−2 (for a≥3 ).

Therefore,