Jacobi polynomials and design theory I

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 $t$ -designs using an operator, known as Aronhold polarization operator. Many authors studied the combinatorial $t$ -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 $t$ -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 $t$ -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 $F_{q}$ be a finite field of order $q$ , where $q$ is a prime power. Then $F_{q}^{n}$ denotes the vector space of dimension $n$ over $F_{q}$ . The elements of $F_{q}^{n}$ are known as vectors. The Hamming weight of $u = (u_{1}, \dots, u_{n}) \in F_{q}^{n}$ is denoted by $wt (u)$ and defined to be the number of $i$ ’s such that $u_{i} \neq 0$ . Let $u = (u_{1}, \dots, u_{n})$ and $v = (v_{1}, \dots, v_{n})$ be the vectors of $F_{q}^{n}$ . Then the inner product of two vectors $u, v \in F_{q}^{n}$ is given by

u \cdot v := u_{1} v_{1} + \dots + u_{n} v_{n} .

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

u \cdot v := u_{1}_{1} + \dots + u_{n}_{n},

where $_{i} := {v_{i}}^{\sqrt{q}}$ . An $F_{q}$ -linear code of length $n$ is a vector subspace of $F_{q}^{n}$ . The elements of an $F_{q}$ -linear code are called codewords. The dual code of an $F_{q}$ -linear code $C$ of length $n$ is defined by

C^{⊥} := {v \in F_{q}^{n} ∣ u \cdot v = 0 for all u \in C} .

An $F_{q}$ -linear code $C$ is called self-dual if $C = C^{⊥}$ . It is well known that the length $n$ of a self-dual code over $F_{q}$ is even and the dimension is $n / 2$ . To study self-dual codes in detail, we refer the readers to [3, 17, 21, 25]. A self-dual code $C$ over $F_{2}$ or $F_{4}$ of length $n \equiv 0 (mod 2)$ having even weight is called Type $I$ and Type $I V$ , respectively. A self-dual code $C$ over $F_{2}$ of length $n \equiv 0 (mod 8)$ is called Type $I I$ if the weight of each codeword of $C$ is multiple of $4$ . Finally, a self-dual code $C$ over $F_{3}$ of length $n \equiv 0 (mod 4)$ is called Type $I I I$ if the weight of each codeword of $C$ is multiple of $3$ .

Definition 2.1.

Let $C$ be an $F_{q}$ -linear code of length $n$ . We denote by $A_{i}^{C}$ the number of codewords in $C$ having Hamming weight $i$ . Then the weight enumerator of $C$ is defined as

W_{C} (x, y) := \sum u \in C x^{n - wt (u)} y^{wt (u)} = n \sum i = 0 A_{i}^{C} x^{n - i} y^{i} .

Definition 2.2.

Let $C$ be an $F_{q}$ -linear code of length $n$ . Then the Jacobi polynomial attached to a set $T$ of coordinate places of the code $C$ is defined as follows:

J_{C, T} (w, z, x, y) := \sum u \in C w^{m_{0} (u)} z^{m_{1} (u)} x^{n_{0} (u)} y^{n_{1} (u)},

where $T \subseteq [n]$ , and $m_{i} (u)$ is the Hamming composition of $u$ on $T$ and $n_{i} (u)$ is the Hamming composition of $u$ on $[n] ∖ T$ .

3. MacWilliams type identity

The MacWilliams type identity for the Jacobi polynomial of an $F_{q}$ –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 $F_{q}$ –linear code with multiple reference vectors.

Definition 3.1.

Let $C$ be an $F_{q}$ –linear code of length $n$ . Then the Jacobi polynomial of $C$ with respect to $ℓ$ reference vectors $w_{1}, \dots, w_{ℓ} \in F_{q}^{n}$ is denoted by $J_{C, w_{1}, \dots, w_{ℓ}} ({x_{a}}_{a \in F_{2}^{ℓ + 1}})$ and defined as

J_{C, w_{1}, \dots, w_{ℓ}} ({x_{a}}_{a \in F_{2}^{ℓ + 1}})

:= \sum u \in C \prod a \in F_{2}^{ℓ + 1} x_{a}^{N_{a} (u, w_{1}, \dots, w_{ℓ})} .

Here we denote by $N_{a} (u_{1}, \dots, u_{g})$ the number of $i$ such that $a = (ϕ (u_{1 i}), \dots, ϕ (u_{g i})) \in F_{2}^{g}$ , where $ϕ (u_{j i}) = 1$ if $u_{j i} \neq 0$ , otherwise $ϕ (u_{j i}) = 0$ .

Note that if $ℓ = 1$ , the above definition is completely equivalent to the Jacobi polynomial with one reference vector (Definition 2.2).

Let $F_{q}$ be a finite field, where $q = p^{f}$ for some prime number $p$ . A character of $F_{q}$ is a homomorphism from the additive group $F_{q}$ to the multiplicative group of non-zero complex numbers. We review [10, 22] to introduce some fixed non-trivial characters over $F_{q}$ . Now let $F (x)$ be a primitive irreducible polynomial of degree $f$ over $F_{p}$ and let $λ$ be a root of $F (x)$ . Then any element $a \in F_{q}$ has a unique representation as:

a = a_{0} + a_{1} λ + a_{2} λ^{2} + \dots + a_{f - 1} λ^{f - 1},

where $a_{i} \in F_{p}$ . For $b \in F_{q}$ , we define $χ_{b} (a) := ζ_{p}^{a_{0} b_{0} + \dots + a_{f - 1} b_{f - 1}}$ , where $ζ_{p}$ is the $p$ -th primitive root $e^{2 π i / p}$ of unity. When $b \neq 0$ , then $χ_{b}$ is a non-trivial character of $F_{q}$ . Let $χ$ be a non-trivial character of $F_{q}$ . Then for any $a \in F_{q}$ , we have the following property:

\sum b \in F_{q} χ (a b) := {\begin{matrix} q & if a = 0, 0 & if a \neq 0. \end{matrix}

Lemma 3.1 ([22]).

Let $C$ be an $F_{q}$ -linear code of length $n$ . For $v \in F_{q}^{n}$ , define

δ_{C^{⊥}} (v) := {\begin{matrix} 1 & if v \in C^{⊥}, 0 & otherwise . \end{matrix}

Then we have the following identity:

δ_{C^{⊥}} (v) = \frac{1}{| C |} \sum u \in C χ (u \cdot v) .

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

Theorem 3.1 (MacWilliams Identity).

Let $C$ be an $F_{q}$ –linear code of length $n$ . Again let $χ$ be a non-trivial character of $F_{q}$ . Let $J_{C, w_{1}, \dots, w_{ℓ}} ({x_{a}}_{a \in F_{2}^{ℓ + 1}})$ be the Jacobi polynomial of $C$ with respect to the reference vectors $w_{1}, \dots, w_{ℓ} \in F_{q}^{n}$ . Then

	$J$	$_{C^{⊥}, w_{1}, \dots, w_{ℓ}} ({x_{a}}_{a \in F_{2}^{ℓ + 1}})$

Proof.

By Lemma 3.1, we can write

	$J$	$_{C^{⊥}, w_{1}, \dots, w_{ℓ}} ({x_{a}}_{a \in F_{2}^{ℓ + 1}})$
		$= \sum u \in C^{⊥} \prod a \in F_{2}^{ℓ + 1} x_{a}^{N_{a} (u, w_{1}, \dots, w_{ℓ})}$

		$= \frac{1}{\| C \|} \sum \begin{matrix} u \in C v \in F_{q}^{n} \end{matrix} χ (u \cdot v) \prod a \in F_{2}^{ℓ + 1} x_{a}^{N_{a} (v, w_{1}, \dots, w_{ℓ})}$
		$= \frac{1}{\| C \|} \sum \begin{matrix} u \in C v \in F_{q}^{n} \end{matrix} χ (u_{1} v_{1} + \dots + u_{n} v_{n}) \prod 1 \leq i \leq n x_{(ϕ (v_{i}), ϕ (w_{1 i}), \dots, ϕ (w_{ℓ i}))}$
		$= \frac{1}{\| C \|} \sum u \in C \prod 1 \leq i \leq n ⎧ ⎨ ⎩ \sum v_{i} \in F_{q} χ (u_{i} v_{i}) x_{(ϕ (v_{i}), ϕ (w_{1 i}), \dots, ϕ (w_{ℓ i}))} ⎫ ⎬ ⎭$
		$= \frac{1}{\| C \|} \sum u \in C \prod a \in F_{q}^{ℓ + 1} {⎛ ⎝ \sum b \in F_{q} χ (a_{1} b) x_{(ϕ (b), ϕ (a_{2}), \dots, ϕ (a_{ℓ + 1}))} ⎞ ⎠}_{1}^{N_{a} (u, w_{1}, \dots, w_{ℓ})}$

Hence the proof is completed. ∎

4. Generalized $t$ -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 $t$ -designs from [7] as follows. Let $t, k, λ$ be the integers such that $λ > 0$ and $k > t > 0$ . Again let $k := (k_{1}, \dots, k_{n})$ such that $k = \sum_{i = 1}^{n} k_{i}$ , $v := (v_{1}, \dots, v_{n})$ such that $v_{i} \geq k_{i}$ for all $i$ . Let $X := (X_{1}, \dots, X_{n})$ , where $X_{i}$ ’s are pairwise disjoint sets with $| X_{i} | = v_{i}$ for all $i$ and

B \subseteq (\frac{X_{1}}{k_{1}}) \times \dots \times (\frac{X_{n}}{k_{n}}) .

Definition 4.1.

A $t$ - $(v, k, λ)$ design or a generalized $t$ -design (in short) is a pair $D := (X, B)$ with the following property: if $t := (t_{1}, \dots, t_{n})$ such that $t = \sum_{i = 1}^{n} t_{i}$ satisfying $0 \leq t_{i} \leq k_{i}$ for all $i$ , then for any choice $T := (T_{1}, \dots, T_{n})$ with $T_{i} \in (\frac{X_{i}}{t_{i}})$ for all $i$ , there are precisely $λ$ members $K := (K_{1}, \dots, K_{n}) \in B$ for which $T_{i} \subseteq K_{i}$ for all $i$ .

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

Let $v = (v_{1}, \dots, v_{ℓ})$ such that $\sum_{i = 1}^{ℓ} v_{i} = n$ and $X = (X_{1}, \dots, X_{ℓ})$ of pairwise disjoint sets $X_{i} \subseteq [n]$ with $| X_{i} | = v_{i}$ . Again let $u = (u_{1}, \dots, u_{n}) \in F_{q}^{n}$ . Then for $X \subseteq [n]$ , we define

	${supp}_{X} (u)$	$:= {i \in X ∣ u_{i} \neq 0},$
	$K (u)$	$:= ({supp}_{X_{1}} (u), \dots, {s u p p}_{X_{ℓ}} (u)),$
	${wt}_{X} (u)$	$:= \| {supp}_{X} (u) \| .$

Again for any positive integer $k$ , let $k = (k_{1}, \dots, k_{ℓ})$ such that $\sum_{i = 1}^{ℓ} k_{i} = k$ . Let $C$ be an $F_{q}$ –linear code of length $n$ . Then

	$C_{k}$	$:= {u \in C ∣ {wt}_{X_{i}} (u) = k_{i} for all i},$
	$B (C_{k})$	$:= {K (u) ∣ u \in C_{k}} .$

In general, $B (C_{k})$ is a multi-set. We say $C_{k}$ is a $t$ - $(v, k, λ)$ design if $(X, B (C_{k}))$ is a $t$ - $(v, k, λ)$ design. We say a code is generalized $t$ -homogeneous if the codewords of every given weight $k$ hold a $t$ - $(v, k, λ)$ 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 $C$ be an $F_{q}$ -linear code of length $n$ . Let $t, k, λ$ be the integers such that $λ > 0$ and $k > t > 0$ . Let $v := (v_{1}, \dots, v_{ℓ})$ such that $v_{1} + \dots + v_{ℓ} = n$ . Let $X := (X_{1}, \dots, X_{ℓ})$ of pairwise disjoint set $X_{1}, \dots, X_{ℓ} \subseteq [n]$ with $| X_{1} | = v_{1}, \dots, | X_{ℓ} | = v_{ℓ}$ . Let $k := (k_{1}, \dots, k_{ℓ})$ such that $k_{1} + \dots + k_{ℓ} = k$ . Then the set of codewords of $C$ form a $t$ - $(v, k, λ)$ design for every given weight $k$ with $t = (t_{1}, \dots, t_{ℓ})$ such $t_{1} + \dots + t_{ℓ} = t$ satisfying $0 \leq t_{i} \leq k_{i}$ for all $i$ if and only if the Jacobi polynomial $J_{C, w_{1}, \dots, w_{ℓ}}$ of $C$ associated to the reference vectors $w_{1}, \dots, w_{ℓ} \in F_{q}^{n}$ such that ${wt}_{X_{i}} (w_{i}) = t_{i} = wt (w_{i})$ for all $i$ , is invariant.

The situation described in the above theorem interprets that the Jacobi polynomial $J_{C, w_{1}, \dots, w_{ℓ}}$ is independent of the choices of the reference vectors $w_{1}, \dots, w_{ℓ} \in F_{q}^{n}$ . In this case, we prefer to denote the Jacobi polynomial as $J_{C, t_{1}, \dots, t_{ℓ}}$ . In particular, when $k = (k)$ and $t = (t)$ , it becomes the Jacobi polynomial $J_{C, t}$ as in [4].

Let $C$ be an $F_{q}$ -linear code of length $n$ . Then the code $C - i$ obtained from $C$ by puncturing at coordinate place $i$ . Now from [22] we have the following lemma.

Lemma 4.1.

Let $C$ be a code of length $n$ . Then

W_{C - i} (x, y) = \frac{1}{n} (\frac{\partial}{\partial x} + \frac{\partial}{\partial y}) W_{C} (x, y) .

Let $P (x_{0}, x_{1})$ be a homogeneous polynomial with indeterminates $x_{0}$ and $x_{1}$ . Again let $P_{x_{0}}^{'}$ (resp. $P_{x_{1}}^{'}$ ) denote the partial derivative with respect to variable $x_{0}$ (resp. $x_{1}$ ). Define the polarization operator $A_{j}$ for any integer $1 \leq j \leq ℓ$ as follows:

(4.1)

A_{j} \cdot P (w_{j}, z_{j}, x_{0}, x_{1}) := w_{j} P_{x_{0}}^{'} (x_{0}, x_{1}) + z_{j} P_{x_{1}}^{'} (x_{0}, x_{1}) .

Here the indeterminates in the above equation denote $x_{a}$ for some $a \in F_{2}^{ℓ + 1}$ as follows: $w_{j} := x_{(0, 0, \dots, 0, 1 (j + 1) th, 0, \dots, 0)}$ , $z_{j} := x_{(1, 0, \dots, 0, 1 (j + 1) th, 0, \dots, 0)}$ , $x_{0} := x_{(0, 0, \dots, 0)}$ , and $x_{1} := x_{(1, 0, \dots, 0)}$ . Now we have a generalization of [4, Theorem 3] as follows.

Theorem 4.2.

Every code $C$ is a generalized $1$ -homogenous if and only if for any $ℓ$ -tuple $(0, \dots, 0, 1, 0, \dots, 0)$ having a single non-zero coordinate, say $j$ -th coordinate with $1$ , we have

(4.2)

J_{C, 0, \dots, 0, 1, 0, \dots, 0} = \frac{1}{n} A_{j} \cdot W_{C} .

Proof.

Let $C$ be generalized $1$ -homogeneous. Then by Lemma 4.1 one can easily find Equation (4.2) is true. Conversely, the hypothesis implies that the Jacobi polynomial $J_{C, 0, \dots, 0, 1, 0, \dots, 0}$ is uniquely determined. Therefore, by Theorem 4.1 we can say that the codewords of every given weight of $C$ form a generalized $1$ -design. Hence $C$ is generalized $1$ -homogeneous. ∎

Theorem 4.3.

If $C$ is generalized $t$ -homogeneous and contains no codeword of weight $\leq t$ then for $t = (t_{1}, \dots, t_{ℓ})$ such that $t_{1} + \dots + t_{ℓ} = t$ we have

J_{C, t_{1}, \dots, t_{ℓ}} = \frac{1}{n (n - 1) \dots (n - t + 1)} A_{ℓ}^{t_{ℓ}} \dots A_{1}^{t_{1}} \cdot W_{C} .

Proof.

The statement is true for $t = 1$ by Theorem 4.2. For $d := (d_{1}, \dots, d_{ℓ})$ such that $d_{1} + \dots + d_{ℓ} = d < t$ satisfying $0 \leq d_{i} \leq t_{i}$ for all $i$ , let us suppose that

J_{C, d_{1}, \dots, d_{ℓ}} = \frac{1}{n (n - 1) \dots (n - d + 1)} A_{ℓ}^{d_{ℓ}} \dots A_{1}^{d_{1}} \cdot W_{C} .

Let $d_{j} < t_{j}$ . Then we have

	$J_{C, d_{1}, \dots, d_{j - 1}, (d_{j} + 1), d_{j + 1}, \dots, d_{ℓ}}$	$= \frac{1}{n - d} A_{j} \cdot J_{C, d_{1}, \dots, d_{j - 1}, d_{j}, d_{j + 1}, \dots, d_{ℓ}}$
		$= \frac{1}{n - d} A_{j} \frac{1}{n (n - 1) \dots (n - d + 1)}$
		$A_{ℓ}^{d_{ℓ}} \dots A_{j + 1}^{d_{j + 1}} A_{j}^{d_{j}} A_{j - 1}^{d_{j - 1}} \dots A_{1}^{d_{1}} \cdot W_{C}$
		$= \frac{1}{n (n - 1) \dots (n - d + 1) (n - d)}$
		$A_{ℓ}^{d_{ℓ}} \dots A_{j + 1}^{d_{j + 1}} A_{j}^{d_{j} + 1} A_{j - 1}^{d_{j - 1}} \dots A_{1}^{d_{1}} \cdot W_{C} .$

The converse implication follows from the proof of Theorem 4.2. ∎

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 $t$ - $(v, k, (λ_{1}^{a_{1}}, \dots, λ_{N}^{a_{N}}))$ is a collection of $k$ -element subsets called blocks of a $v$ -element set (the varieties) and a partition of the set of all $t$ -tuples into $N$ groups such that every $t$ -set belonging to the $i^{th}$ group (comprising $a_{i}$ such $t$ -sets) is contained in exactly $λ_{i}$ blocks. Notice that for $N = 1$ the design coincide with a $t$ -design. A packing (resp. covering) design with parameters $t$ - $(v, k, λ)$ is a design with ${max}_{i} (λ_{i}) = λ$ (resp. ${min}_{i} (λ_{i}) = λ$ ). The maximum (resp. minimum) number of blocks of a packing (resp. covering) design denoted by $D_{λ} (v, k, t)$ (resp. $C_{λ} (v, k, t)$ ).

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 $G$ be a finite subgroup of $G L (2, C)$ , and $G$ acts on a polynomial ring of two variables, say $x, y$ . Then it is well-known from [28, Theorem 1] that the classical Molien series gives the linearly independent homogeneous invariant polynomials of $G$ . Later, R.P. Stanley [29] introduce the notion of bivariate Molien series that computes invariant polynomials of $G$ by their homogeneous degrees in $w, z$ and $x, y$ . R.P. Stanley [29] defined the bivariate Molien series as follows:

f (u, v) := \frac{1}{| G |} \sum g \in G \frac{1}{det (1 - u g) det (1 - v g)} .

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 $G$

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

d

f (u, v)

f [d]

In the following examples, we study two types of codes over $F_{q}$ ; Type III and Type IV, that hold $t$ -designs with parameters $t$ - $(v, k, λ)$ , and we would like to give an upper (resp. lower) bound of $D_{λ} (v, k, t)$ (resp. $C_{λ} (v, k, t)$ ) of a packing (resp. covering) design corresponding to the parameters. To do so, firstly, we compute the homogeneous part $f [d]$ of $f (u, v)$ corresponding to a code of length $d$ . The coefficients of $f [d]$ 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 $λ$ ’s of $t$ - $(n, k, λ_{1}^{a_{1}}, \dots, λ_{N}^{a_{N}})$ . Finally, the coefficient of the term $x^{n - k} y^{k}$ in the weight enumerator of the code obtains the upper (resp. lower) bound of $D_{λ} (v, k, t)$ (resp. $C_{λ} (v, k, t)$ ). Note that a packing (resp. covering) design is a simple design.

5.1. Type III codes

The MacWilliams identity and the modulo $3$ congruence condition yield that the weight enumerator of a Type III code remains invariant under the action of group $G_{3}$ of order $48$ which is generated by the following two matrices:

\frac{1}{\sqrt{3}} [\begin{matrix} 1 & 2 1 & - 1 \end{matrix}] and [\begin{matrix} 1 & 0 0 & e^{2 π i / 3} \end{matrix}] .

For the case of the group $G_{3}$ , we get from the Magma computations that the denominator of $f (u, v)$ is in the form of $d (u) d (v)$ , where

d (u) = (u - 1)^{2} (u + 1)^{2} (u^{2} + 1)^{2} (u^{2} - u + 1) (u^{2} + u + 1) (u^{4} - u^{2} + 1) .

Example 5.1 (length $4$ ).

Let $C_{4}^{III}$ be a ternary self-dual code of length $4$ in [18]. Then

f [4] = u^{4} + u^{3} v + u^{2} v^{2} + u v^{3} + v^{4} .

Since $C_{4}^{III}$ holds $3$ -design, we assume that $| T | = 1, 2, 3$ . Then

	$J_{C_{4}^{III}, 1}$	$= \frac{1}{4} A W_{C_{4}^{III}} (x, y)$
		$= w (x^{3} + 2 y^{3}) + 6 z x y^{2},$
	$J_{C_{4}^{III}, 2}$	$= \frac{1}{4 \cdot 3} A^{2} W_{C_{4}^{III}} (x, y)$
		$= w^{2} x^{2} + 4 w z y^{2} + 4 z^{2} x y,$
	$J_{C_{4}^{III}, 3}$	$= \frac{1}{4 \cdot 3 \cdot 2} A^{3} W_{C_{4}^{III}} (x, y)$
		$= w^{3} x + 6 w z^{2} y + 2 z^{3} x .$

By dividing the coefficient of the term $z^{t} y^{3 - t}$ ( $t = 1, 2, 3$ ) in the Jacobi polynomial by $2$ , we obtain the values of $λ$ . Since the coefficient of the term $u^{4 - t} v^{t}$ in $f [d]$ is $1$ , we obtain the group divisible design $t$ - $(4, 3, λ)$ . Then the maximum number of blocks of $t$ - $(4, 3, λ)$ design is 4, and the minimum number of blocks of $t$ - $(4, 3, λ)$ design is 4. Therefore, $D_{λ} (4, 3, t) \leq 4 \leq C_{λ} (4, 3, t)$ .

Example 5.2 (length $8$ ).

Let $C_{8}^{III}$ be a ternary self-dual code of length $8$ in [18]. Then

f [8] = u^{8} + u^{7} v + 2 u^{6} v^{2} + 2 u^{5} v^{3} + 2 u^{4} v^{4} + 2 u^{3} v^{5} + 2 u^{2} v^{6} + u v^{7} + v^{8} .

Since $C_{8}^{III}$ holds $1$ -design, we assume that $| T | = 1$ . Then

J_{C_{8}^{III}, 1} = \frac{1}{8} A W_{C_{8}^{III}} (x, y) = w (x^{7} + 10 x^{4} y^{3} + 16 x y^{6}) + z (6 x^{5} y^{2} + 48 x^{2} y^{5}) .

The space of Jacobi polynomials $J_{C_{8}^{III}, T}$ with $| T | = 2$ may be generated by the two polynomials

	$J_{C_{8}^{III}, 2}^{1}$	$= w^{2} (x^{6} + 8 x^{3} y^{3}) + w z (4 x^{4} y^{2} + 32 x y^{5}) + z^{2} (4 x^{5} y + 32 x^{2} y^{4}),$
	$J_{C_{8}^{III}, 2}^{2}$	$= w^{2} (4 x^{3} y^{3} + 4 y^{6}) + w z (12 x^{4} y^{2} + 24 x y^{5}) + 36 z^{2} x^{2} y^{4} .$

Combining these two equations, we obtain $2$ -designs with parameters

	$2$	$- (8, 3, (2^{12}, 0^{16})),$
	$2$	$- (8, 6, (8^{12}, 9^{16})) .$

Since $k = 3 ℓ$ ( $1 \leq ℓ \leq 2$ ), dividing the coefficient of the term $z^{t} y^{k - t}$ ( $t = 2, 3$ ) in the Jacobi polynomials by $2^{ℓ}$ , we obtain the values of $λ_{1}, λ_{2}$ . Dividing the coefficient of the term $x^{8 - k} y^{k}$ in the weight enumerator of the code by $2^{ℓ}$ , we obtain an upper (resp. lower) bound of $D_{λ} (8, k, t)$ (resp. $C_{λ} (8, k, t)$ ).

	$D_{2} (8, 3, 2)$	$\leq 8 \leq C_{0} (8, 3, 2),$
	$D_{9} (8, 6, 2)$	$\leq 16 \leq C_{8} (8, 6, 2) .$

The space of Jacobi polynomials $J_{C_{8}^{III}, T}$ with $| T | = 3$ may be generated by the two polynomials

	$J_{C_{8}^{III}, 3}^{1}$	$= w^{3} (x^{5} + 8 x^{2} y^{3}) + w z^{2} (6 x^{4} y + 48 x y^{4}) + z^{3} (2 x^{5} + 16 x^{2} y^{3}),$
	$J_{C_{8}^{III}, 3}^{2}$	$= w^{3} (x^{5} + 2 x^{2} y^{3}) + w^{2} z (10 x^{3} y^{2} + 8 y^{5})$
		$+ w z^{2} (4 x^{4} y + 32 x y^{4}) + 24 z^{3} x^{2} y^{3},$

which gives packing and covering designs

	$D_{1} (8, 3, 3)$	$\leq 8 \leq C_{0} (8, 3, 3),$
	$D_{6} (8, 6, 3)$	$\leq 16 \leq C_{4} (8, 6, 3) .$

Example 5.3 (length $12$ ).

Let $C_{12}^{III}$ be the first ternary self-dual code of length $12$ in [18].

f [12] = 2 u^{12} + 2 u^{11} v + 3 u^{10} v^{2} + 4 u^{9} v^{3} + 4 u^{8} v^{4} + 4 u^{7} v^{5} + 5 u^{6} v^{6} + \dots .

Since $C_{12}^{III}$ holds $5$ -design, we observe that

	$J_{C_{12}^{III}, 1}$	$= \frac{1}{12} A W_{C_{12}^{III}} (x, y)$
		$= w (x^{11} + 132 x^{5} y^{6} + 110 x^{2} y^{9}) + z (132 x^{6} y^{5} + 330 x^{3} y^{8} + 24 y^{11}),$
	$J_{C_{12}^{III}, 2}$	$= \frac{1}{12 \cdot 11} A^{2} W_{C_{12}^{III}} (x, y)$
		$= w^{2} (x^{10} + 60 x^{4} y^{6} + 20 x y^{9}) + 2 w z (72 x^{5} y^{5} + 90 x^{2} y^{8})$
		$+ z^{2} (60 x^{6} y^{4} + 240 x^{3} y^{7} + 24 y^{10}),$
	$J_{C_{12}^{III}, 3}$	$= \frac{1}{12 \cdot 11 \cdot 10} A^{3} W_{C_{12}^{III}} (x, y)$
		$= w^{3} (x^{9} + 24 x^{3} y^{6} + 2 y^{9}) + w^{2} z (108 x^{4} y^{5} + 54 x y^{8})$
		$+ w z^{2} (108 x^{5} y^{4} + 216 x^{2} y^{7})$
		$+ z^{3} (24 x^{6} y^{3} + 168 x^{3} y^{6} + 24 y^{9}),$
	$J_{C_{12}^{III}, 4}$	$= \frac{1}{12 \cdot 11 \cdot 10 \cdot 9} A^{4} W_{C_{12}^{III}} (x, y)$
		$= w^{4} (x^{8} + 8 x^{2} y^{6}) + w^{3} z (64 x^{3} y^{5} + 8 y^{8}) + w^{2} z^{2} (120 x^{4} y^{4} + 96 x y^{7})$
		$+ w z^{3} (64 x^{5} y^{3} + 224 x^{2} y^{6}) + z^{4} (8 x^{6} y^{2} + 112 x^{3} y^{5} + 24 y^{8}),$
	$J_{C_{12}^{III}, 5}$	$= \frac{1}{12 \cdot 11 \cdot 10 \cdot 9 \cdot 8} A^{5} w_{C_{12}^{% III}} (x, y)$
		$= w^{5} (x^{7} + 2 x y^{6}) + 30 w^{4} z x^{2} y^{5} + w^{3} z^{2} (100 x^{3} y^{4} + 20 y^{7})$
		$+ w z^{4} (30 x^{5} y^{2} + 210 x^{2} y^{5}) + w^{2} z^{3} (100 x^{4} y^{3} + 140 x y^{6})$
		$+ z^{5} (2 x^{6} y + 70 x^{3} y^{4} + 24 y^{7}) .$

The space of Jacobi polynomials $J_{C_{12}^{III}, T}$ with $| T | = 6$ is generated by the two polynomials

J_{C_{12}^{III}, 6}^{1}

= w^{6} (x^{6} + 2 y^{6}) + 90 w^{4} z^{2} x^{2} y^{4} + w^{3} z

	$J$	$_{C^{⊥}, w_{1}, \dots, w_{ℓ}} ({x_{a}}_{a \in F_{2}^{ℓ + 1}})$
		$= \sum u \in C^{⊥} \prod a \in F_{2}^{ℓ + 1} x_{a}^{N_{a} (u, w_{1}, \dots, w_{ℓ})}$

		$= \frac{1}{\| C \|} \sum \begin{matrix} u \in C v \in F_{q}^{n} \end{matrix} χ (u \cdot v) \prod a \in F_{2}^{ℓ + 1} x_{a}^{N_{a} (v, w_{1}, \dots, w_{ℓ})}$
		$= \frac{1}{\| C \|} \sum \begin{matrix} u \in C v \in F_{q}^{n} \end{matrix} χ (u_{1} v_{1} + \dots + u_{n} v_{n}) \prod 1 \leq i \leq n x_{(ϕ (v_{i}), ϕ (w_{1 i}), \dots, ϕ (w_{ℓ i}))}$
		$= \frac{1}{\| C \|} \sum u \in C \prod 1 \leq i \leq n ⎧ ⎨ ⎩ \sum v_{i} \in F_{q} χ (u_{i} v_{i}) x_{(ϕ (v_{i}), ϕ (w_{1 i}), \dots, ϕ (w_{ℓ i}))} ⎫ ⎬ ⎭$
		$= \frac{1}{\| C \|} \sum u \in C \prod a \in F_{q}^{ℓ + 1} {⎛ ⎝ \sum b \in F_{q} χ (a_{1} b) x_{(ϕ (b), ϕ (a_{2}), \dots, ϕ (a_{ℓ + 1}))} ⎞ ⎠}_{1}^{N_{a} (u, w_{1}, \dots, w_{ℓ})}$

Jacobi polynomials and design theory I

Harmonic Tutte polynomials of matroids II

There are EXACTLY 1493804444499093354916284290188948031229880469556 Ways to Derange a Standard Deck of Cards (ignoring suits) [and many other such useful facts]

Residues of skew rational functions and linearized Goppa codes

Planar Polynomials arising from Linearized polynomials

New Julia and Mandelbrot Sets for Jungck Ishikawa Iterates

A Formula for Type III Sums of Squares

A Cartesian Bicategory of Polynomial Functors in Homotopy Type Theory

1. Introduction

2. Preliminaries

Definition 2.1.

Definition 2.2.

3. MacWilliams type identity

Definition 3.1.

Lemma 3.1 ([22]).

Theorem 3.1 (MacWilliams Identity).

Proof.

4. Generalized $t$ -designs and Jacobi polynomials

Definition 4.1.

Theorem 4.1.

Lemma 4.1.

Theorem 4.2.

Proof.

Theorem 4.3.

Proof.

5. Designs and Molien series

5.1. Type III codes

Example 5.1 (length $4$ ).

Example 5.2 (length $8$ ).

Example 5.3 (length $12$ ).

Jacobi polynomials and design theory I

Related Research

Harmonic Tutte polynomials of matroids II

There are EXACTLY 1493804444499093354916284290188948031229880469556 Ways to Derange a Standard Deck of Cards (ignoring suits) [and many other such useful facts]

Residues of skew rational functions and linearized Goppa codes

Planar Polynomials arising from Linearized polynomials

New Julia and Mandelbrot Sets for Jungck Ishikawa Iterates

A Formula for Type III Sums of Squares

A Cartesian Bicategory of Polynomial Functors in Homotopy Type Theory

1. Introduction

2. Preliminaries

Definition 2.1.

Definition 2.2.

3. MacWilliams type identity

Definition 3.1.

Lemma 3.1 ([22]).

Theorem 3.1 (MacWilliams Identity).

Proof.

4. Generalized t-designs and Jacobi polynomials

Definition 4.1.

Theorem 4.1.

Lemma 4.1.

Theorem 4.2.

Proof.

Theorem 4.3.

Proof.

5. Designs and Molien series

5.1. Type III codes

Example 5.1 (length 4).

Example 5.2 (length 8).

Example 5.3 (length 12).

4. Generalized $t$ -designs and Jacobi polynomials

Example 5.1 (length $4$ ).

Example 5.2 (length $8$ ).

Example 5.3 (length $12$ ).