## 1. Introduction

A. Bonnecaze et al. [4] took the notion of Jacobi polynomials, a celebrated generalization of weight enumerators [20, 22] that were introduced by M. Ozeki [26] for codes as an analogue to Jacobi forms [2, 16] as a powerful generalization of modular form [15, 27] of Lattices [8]. They gave a formula to compute the Jacobi polynomials of a binary code as an application of combinatorial -designs using an operator, known as Aronhold polarization operator. Many authors studied the combinatorial -designs and discussed their properties in [1, 14, 23, 24] that were derived from codes and their analogies. Moreover, P.J. Cameron [7] gave the notion of generalized -designs and discussed its properties. Furthermore, A. Bonnecaze et al. [4] constructed various types of designs such as group divisible designs, packing designs and covering designs. To establish the relationship between these designs and the Jacobi polynomials, they studied Jacobi polynomials for Type II codes through invariant theory [17, 25].

In this paper, we give the generalizations and analogues of some results in [4]. We define the Jacobi polynomials with multiple reference vectors for codes, and give the MacWilliams type identity for it. As an analogue of the combinatorial interpretation of the polarization that was given in [4], is given here for codes that holds generalized -designs for every given weight of the codewords. In addition, we study some Type III (resp. Type IV) codes of specific lengths, and determine the polynomials that generate the space of Jacobi polynomials for a Type III (resp. Type IV) code with respect to reference vectors of a particular length. Moreover, we observe from the examples that the number of blocks of a packing (resp. covering) design correspond to the coefficients in Jacobi polynomials.

This paper is organized as follows. In Section 2, we discuss the basic definitions and properties of codes that needed to understand this paper. In Section 3, we give the MacWilliams type identity (Theorem 3.1) for the Jacobi polynomials of a code with multiple reference vectors. In Section 4, we see how polarization operator acts to obtain the Jacobi polynomials with multiple reference vectors (Theorem 4.2, Theorem 4.3). In Section 5, we disclose some facts between a Type III (resp. Type IV) code of specific length and designs of various kinds with the help of the Jacobi polynomials. Finally, we conclude the paper with some remarks in Section 6.

All computer calculations in this paper were done with the help of Magma [6].

## 2. Preliminaries

Let be a finite field of order ,
where is a prime power.
Then denotes the vector space of dimension over .
The elements of are known as *vectors*.
The *Hamming weight* of
is denoted by and
defined to be the number of ’s such that .
Let
and
be the vectors of .
Then the *inner product* of two vectors
is given by

If is an even power of an arbitrary prime , then it is convenient to consider another inner product given by

where .
An -linear code of length is a vector subspace of .
The elements of an -linear code are called *codewords*.
The *dual code* of an -linear code
of length is defined by

An -linear code is called *self-dual* if .
It is well known that
the length of a self-dual code over is even and the dimension is .
To study self-dual codes in detail, we refer the readers
to [3, 17, 21, 25].
A self-dual code over or of length having even weight is called *Type* and *Type* , respectively. A self-dual code over of length is called *Type* if the weight of each codeword of is multiple of . Finally, a self-dual code over of length is called *Type* if the weight of each codeword of is multiple of .

###### Definition 2.1.

Let be an -linear code of length .
We denote by the number of codewords in having
Hamming weight .
Then the *weight enumerator* of
is defined as

###### Definition 2.2.

Let be an -linear code of length . Then
the *Jacobi polynomial* attached to a set of
coordinate places of the code is defined as follows:

where , and is the Hamming composition of on and is the Hamming composition of on .

## 3. MacWilliams type identity

The MacWilliams type identity for the Jacobi polynomial of an –linear code with one reference vector was given in [26]. In this section, we give the MacWilliams type identity for the Jacobi polynomial of an –linear code with multiple reference vectors.

###### Definition 3.1.

Let be an –linear code of length .
Then the *Jacobi polynomial* of
with respect to reference vectors
is denoted by
and
defined as

Here we denote by the number of such that , where if , otherwise .

Note that if , the above definition is completely equivalent to the Jacobi polynomial with one reference vector (Definition 2.2).

Let be a finite field, where for some prime number .
A *character* of is a homomorphism from the additive group
to the multiplicative group of non-zero complex numbers.
We review [10, 22] to introduce some fixed non-trivial characters over .
Now let be a primitive irreducible polynomial
of degree over and let be a root of .
Then any element has a unique representation as:

where . For , we define , where is the -th primitive root of unity. When , then is a non-trivial character of . Let be a non-trivial character of . Then for any , we have the following property:

###### Lemma 3.1 ([22]).

Let be an -linear code of length . For , define

Then we have the following identity:

Now we give the MacWilliams type identity for the Jacobi polynomial of an –linear code with respect to multiple reference vectors.

###### Theorem 3.1 (MacWilliams Identity).

Let be an –linear code of length . Again let be a non-trivial character of . Let be the Jacobi polynomial of with respect to the reference vectors . Then

## 4. Generalized -designs and Jacobi polynomials

Bonnecaze et al. introduced an operator called *polarization operator* in [4],
and using this operator, they gave a formula to evaluate the Jacobi polynomial of a binary code from the weight enumerator of the code.
In this section, we give a generalized form of the polarization operation,
and present an application of this operator in the evaluation of the Jacobi polynomial
of a non-binary code associated to the multiple reference vectors.

First we recall the definition of generalized -designs from [7] as follows. Let be the integers such that and . Again let such that , such that for all . Let , where ’s are pairwise disjoint sets with for all and

###### Definition 4.1.

A - *design* or a *generalized* -*design* (in short) is a pair
with the following property:
if such that satisfying for all ,
then for any choice with for all ,
there are precisely members for which for all .

Note that in the case when and ,
this is precisely the definition of a combinatorial
- design or a -*design* (in short).
We can construct the generalized -designs from codes as follows.

Let such that and of pairwise disjoint sets with . Again let . Then for , we define

Again for any positive integer , let such that . Let be an –linear code of length . Then

In general, is a multi-set. We say is a - design if is a - design. We say a code is *generalized* -*homogeneous* if the codewords of every given weight hold a
- design.

From the above discussion we have the following result. We omit the proof of the theorem since it follows from the above definitions.

###### Theorem 4.1.

Let be an -linear code of length . Let be the integers such that and . Let such that . Let of pairwise disjoint set with . Let such that . Then the set of codewords of form a - design for every given weight with such satisfying for all if and only if the Jacobi polynomial of associated to the reference vectors such that for all , is invariant.

The situation described in the above theorem interprets that the Jacobi polynomial is independent of the choices of the reference vectors . In this case, we prefer to denote the Jacobi polynomial as . In particular, when and , it becomes the Jacobi polynomial as in [4].

Let be an -linear code of length .
Then the code obtained from by *puncturing* at coordinate place .
Now from [22] we have the following lemma.

###### Lemma 4.1.

Let be a code of length . Then

Let be a homogeneous polynomial with indeterminates and . Again let (resp. ) denote the partial derivative with respect to variable (resp. ). Define the polarization operator for any integer as follows:

(4.1) |

Here the indeterminates in the above equation denote for some as follows: , , , and . Now we have a generalization of [4, Theorem 3] as follows.

###### Theorem 4.2.

Every code is a generalized -homogenous if and only if for any -tuple having a single non-zero coordinate, say -th coordinate with , we have

(4.2) |

###### Proof.

Let be generalized -homogeneous. Then by Lemma 4.1 one can easily find Equation (4.2) is true. Conversely, the hypothesis implies that the Jacobi polynomial is uniquely determined. Therefore, by Theorem 4.1 we can say that the codewords of every given weight of form a generalized -design. Hence is generalized -homogeneous. ∎

###### Theorem 4.3.

If is generalized -homogeneous and contains no codeword of weight then for such that we have

## 5. Designs and Molien series

Bonnecaze et al. [4] studied certain length of Type II codes to focus some relation between Jacobi polynomials and designs. In this section, we follow the idea, and establish the connection between Jacobi polynomials and designs for some Type III and Type IV codes. We would like to mention that in this section, we study Jacobi polynomials with one reference vector. To overcome all sorts of confusions, we refer the readers to [4] for notations and symbols.

First, let us recall [4] for the definitions of various types of designs.
A *design* with parameters - is a collection of -element subsets called *blocks* of a -element set (the *varieties*) and a partition of the set of all -tuples into *groups* such that every -set belonging to the group (comprising such -sets) is contained in exactly blocks.
Notice that for the design coincide with a -design.
A *packing* (resp. *covering*) *design* with parameters - is a design with
(resp. ).
The maximum (resp. minimum) number of blocks of a packing (resp. covering) design denoted by (resp. ).

The study of weight polynomials of a code with the help of invariant theory
is a very convenient and powerful technique, as shown in [25, 28].
Let be a finite subgroup of , and
acts on a polynomial ring of two variables, say .
Then it is well-known from [28, Theorem 1] that
the classical Molien series
gives the linearly independent homogeneous invariant polynomials of .
Later, R.P. Stanley [29] introduce the
notion of *bivariate* *Molien series* that computes invariant polynomials
of by their homogeneous degrees in and .
R.P. Stanley [29] defined
the bivariate Molien series as follows:

A. Bonnecaze et al. [4] showed that the bivariate Molien series plays an important role to describe the relation between the Jacobi polynomials of codes and designs such as group divisible designs, packing (resp. covering) designs. To compute all the invariant polynomials of

explicitly, it is convenient to classify the invariants by their degrees. We denote the homogeneous part of degree

of by .In the following examples, we study two types of codes over ; Type III and Type IV, that hold -designs with parameters -, and we would like to give an upper (resp. lower) bound of (resp. ) of a packing (resp. covering) design corresponding to the parameters. To do so, firstly, we compute the homogeneous part of corresponding to a code of length . The coefficients of determine the number of polynomials that are needed to generate the space of Jacobi polynomials corresponding to the reference sets with a particular cardinality. The number of those Jacobi polynomials determines the number of ’s of -. Finally, the coefficient of the term in the weight enumerator of the code obtains the upper (resp. lower) bound of (resp. ). Note that a packing (resp. covering) design is a simple design.

### 5.1. Type III codes

The MacWilliams identity and the modulo congruence condition yield that the weight enumerator of a Type III code remains invariant under the action of group of order which is generated by the following two matrices:

For the case of the group , we get from the Magma computations that the denominator of is in the form of , where

###### Example 5.1 (length ).

By dividing the coefficient of the term () in the Jacobi polynomial by , we obtain the values of . Since the coefficient of the term in is , we obtain the group divisible design -. Then the maximum number of blocks of - design is 4, and the minimum number of blocks of - design is 4. Therefore, .

###### Example 5.2 (length ).

Let be a ternary self-dual code of length in [18]. Then

Since holds -design, we assume that . Then

The space of Jacobi polynomials with may be generated by the two polynomials

Combining these two equations, we obtain -designs with parameters

Since (), dividing the coefficient of the term () in the Jacobi polynomials by , we obtain the values of . Dividing the coefficient of the term in the weight enumerator of the code by , we obtain an upper (resp. lower) bound of (resp. ).

The space of Jacobi polynomials with may be generated by the two polynomials

which gives packing and covering designs

###### Example 5.3 (length ).

Let be the first ternary self-dual code of length in [18].

Since holds -design, we observe that

The space of Jacobi polynomials with is generated by the two polynomials