The support designs of the triply even codes of length
Abstract
A triply even code is a binary linear code in which the weight of every codeword is divisible by . The triply even codes of length have been classified by Betsumiya and Munemasa. Herein we study the support designs of triply even codes of length and present the complete list of triply even codes of length with the support designs obtained from the Assmus–Mattson theorem. Moreover, we show that some of such codes have the support designs. This is the first example of a code having the support designs for all weights obtained from the Assmus–Mattson theorem and has the support designs for some weight with some .
Key Words and Phrases.
support designs, triply even codes, harmonic weight enumerator.
2010 Mathematics Subject Classification.
Primary 05B05;
Secondary 94B05, 20B25.
1 Introduction
Let be the finite field of elements. A binary linear code of length is a subspace of . For , the weight of x is defined as follows: The minimum distance of a code is
A linear code of length , dimension , and minimum distance is called an code (or code for short).
A  design (or design for short) is a pair , where is a set of points of cardinality , and a collection of element subsets of called blocks, with the property that any points are contained in precisely blocks.
The support of a nonzero vector , is the set of indices of its nonzero coordinates: . The support design of a code of length for a given nonzero weight is the design with points of coordinate indices, and blocks the supports of all codewords of weight .
A triply even code is a binary linear code in which the weight of every codeword is divisible by ; they have previously been classified by Betsumiya and Munemasa [5, 4]. Herein we study the support designs of triply even codes of length and present the complete list of triply even codes of length which have the support designs obtained from the Assmus–Mattson theorem. It is interesting to note that some of such codes have the support designs. This is the first example of code having the support designs for all weights obtained from the Assmus–Mattson theorem and has the support designs for some weights with some .
This paper is organized as follows. In Section , we review the concept of harmonic weight enumerators which are used in the proof of the main results. In Section , we present the main result, namely, we study the support designs of the triply even codes of length . Finally, in Section , we give some remarks.
2 The harmonic weight enumerators
In this section, we review the concept of the harmonic weight enumerators.
Let be a code of length . The weight distribution of a code is the sequence , where is the number of codewords of weight . The polynomial
is called the weight enumerator of . The weight enumerator of a code and its dual are related. The following theorem, due to MacWilliams, is called the MacWilliams identity:
Theorem 2.1 ([6]).
Let be the weight enumerator of an code over and let be the weight enumerator of the dual code . Then
A striking generalization of the MacWilliams identity was obtained by Bachoc [2], who gave the concept of harmonic weight enumerators and a generalization of the MacWilliams identity. The harmonic weight enumerators have many applications; particularly, the relations between coding theory and design theory are reinterpreted and progressed by the harmonic weight enumerators [2, 3]. For the reader’s convenience, we quote the definitions and properties of discrete harmonic functions from [2, 9].
Let be a finite set (which will be the set of coordinates of the code) and let be the set of its subsets, while, for all , is the set of its subsets. We denote by , the free real vector spaces spanned by respectively the elements of , . An element of is denoted by
and is identified with the realvalued function on given by .
Such an element can be extended to an element by setting, for all ,
If an element is equal to some , for , we say that has degree . The differentiation is the operator defined by the linear form
for all and for all , and is the kernel of :
Theorem 2.2 ([9]).
A set of blocks is a design if and only if for all , .
In [2], the harmonic weight enumerator associated with a binary linear code was defined as follows:
Definition 2.3.
Let be a binary code of length and let . The harmonic weight enumerator associated with and is
Bachoc proved the following MacWilliamstype equality:
Theorem 2.4 ([2]).
Let be the harmonic weight enumerator associated with the code and the harmonic function of degree . Then
where is a homogeneous polynomial of degree , and satisfies
3 The support designs of triply even codes of length
In this section, we study the designs of triple even codes of length .
The following theorem is due to Assmus and Mattson [1]. It is one of the most important theorems in coding theory and design theory:
Theorem 3.1 ([1]).
Let be an linear code over and be the dual code. Denote by the largest integer such that , and define similarly for the dual code . Suppose that for some integer , there are at most nonzero weights in such that . Then:

the support design for any weight , in is a design;

the support design for any weight , in is a design.
If a design () is obtained from some linear code by this theorem, the code is said to be applicable to the Assmus–Mattson theorem.
In [4, 5], triply even codes of length 48 are classified where each code is described by the form . In this paper, we use the form .
Proposition 3.2.
If a triply even code length is applicable to the Assmus–Mattson theorem, then one of the following:

, ,
,
,
, , . 
Proof.
Let be a triply even code length in . Then have weight 0,16,24,32,48 and the dual code of have the minimum weight . Since there are nonzero weights with in , we can take .
In the case , these codes have weight 0, 24, 48 and the dual codes have the minimum weight 2. Since there are nonzero weights with , we can take .
∎
The following Lemma is easily seen:
Lemma 3.3 ([8, Page 3, Proposition 1.4]).

Let be the block set of a  design. Then is divisible by .

Let be the block set of a  design. Then is divisible by .
We now present the main result:
Theorem 3.4.
Let be a triply even code length in Proposition 3.2. Let and be the support design of weight of and .

For all , and are designs.

If is a code in Proposition 3.2 (A) except for , is a design but is not a design. For the other cases, and are not designs.
Proof.
By Proposition 3.2, we have .
Next, we show that if is a code according to Proposition 3.2 (A) except for , is a design but is not a design. Let be the harmonic weight enumerator associated with the code in Proposition 3.2 (A) and the harmonic function of degree . Then we have
where and are not equal to .
By Theorem 2.4, there exists coefficients such that
Since has minimum weight , the coefficient of in is equal to . Then we have . Hence we have
By a direct computation, the coefficient of in is equal to . Hence is a design. We have checked numerically that the number of blocks of is not divisible by . Therefore, is not a design by Lemma 3.3 (2).
In the case , this code has the weight enumerator
By Theorem 2.1, we have , where is the number of weight of the dual code. Hence there are no blocks of .
Next, we show that for the other cases, and are not designs. We have checked numerically that the number of the blocks of and except for is not divisible by . Therefore, and except for are not designs by Lemma 3.3 (1).
Let be a triply even code of length in Proposition 3.2 (B). If and is the support design for all weight of and , we have checked numerically that the number of the blocks of and is not divisible by . Therefore, and are not designs by Lemma 3.3 (1). The numbers of the blocks are listed in one of the author’s homepage [12].
This completes the proof of Theorem 3.4. ∎
Remark 3.5.
Three codes , and are not applicable to the Assmus–Mattson theorem, but their support designs for all weight are designs since their codes are generated by the minimum weights which divide 48 coordinates into equal parts. Their codes have a transitive automorphism group.
4 designs from triple even codes of length
We list designs obtained from Theorem 3.4 in Table 1. In this section, we give the concluding remarks related to designs of triple even codes of length discussed in Section 3.
Remark 4.1.
It is interesting to note that the dual code of the first triply even code is called Miyamoto’s moonshine code [13]. This triply even code has the weight enumerator
Using Theorem 2.1, we obtained the weight enumerator of the dual code. Then we have , where is the number of weight of the dual code. By Theorem 3.4, is a  design.
In the case dimension of , there are five triply even codes [129,130,131,132,133]. We have checked by Magma [7] that their codes give the five non isomorphic  designs. Similarly, each triply even code in dimension – gives a different design.
In the case of triply even code , there are no code words of weight of the dual code.
Remark 4.2.
Remark 4.3.
We have checked by Magma [7] that for the design obtained from Theorem 3.4 in Table 1, the codewords of weight generate the code . This gives rise to a natural problem:
Problem 4.4.
Let be a linear code. Characterize the weight such that the codewords of weight generate the code . Moreover, do we characterize such weight from the point of view of design theory?
Remark 4.5.
One of the main result of the present paper is to give the first example that a code has the support designs for all weights obtained from the Assmus–Mattson theorem and has the support designs for some of the weights with some . We conclude the present paper to with the following problem:
Problem 4.6.
For , find an example that a code has the support designs for all weight obtained from the Assmus–Mattson theorem and has the support designs for some weight with some .
Dimension  [Code Id]   

Weight distribution for  Numbers  
7  [144]   
1  
8 
[129,130,131,132,133]   
5  
9 
[59,60,61,62,63,64,65,66,67,68,69,1109,1712,1714,1716,1960]   
16  
10 
[16,17,18,19,20,21,22,549,550,554,1001,1245,1246,1247]   
14  
11 
[6,7,154,520]   
4  
12 
[3]   
1  
13 
[1]   
0 
Acknowledgments
The authors thank Koichi Betsumiya and Akihiro Munemasa for his helpful discussions and computations on this research. The first author is supported by JSPS KAKENHI (15K04775, 17K05164, 18K03217).
References
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[11]
T. Miezaki and H. Nakasora,
An upper bound of the value of of the support designs of extremal binary doubly even selfdual codes, Des. Codes Cryptogr., 79 (2016), 3746. 
[12]
T. Miezaki,
Tsuyoshi Miezaki’s website: https://sites.google.com/site/tmiezaki/data  [13] M. Miyamoto, A new construction of the Moonshine vertex operator algebras over the real number field, Ann. of Math., 159 (2004), 535–596.