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On Lee Association Scheme over $\mathbb{Z}_4$, Terwilliger algebras and the Assmus-Mattson Theorem (Research on finite groups, algebraic combinatorics and vertex operator algebras)

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(1)68. 数理解析研究所講究録 第2053巻 2017年 68-79. On Lee Association Scheme. over. \mathb {Z}_{4} Terwilliger algebras ,. and the Assmus‐Mattson Theorem. *. John Vincent S. Morales $\dag er$. Abstract Let C denote. corresponding. linear code of. a. length. n over a. finite field. $\Gamma$_{q}. and let. C^{\perp}. denote the. dual. The Assmus‐Mattson theorem states that combinatorial. designs. be obtained from the supports of codewords of C with fixed weight type when‐ ever the Hamming weight enumerators of C and C^{\perp} satisfy certain conditions. This can. famous result has been. and extended to many different. strengthened. settings including. the Assmus‐Mattson type theorems for \mathb {Z}_{4} ‐linear codes due to Tanabe (2003), and due to Shin, Kumar and Helleseth (2004). In this paper, we discuss an Assmus‐Mattson. type theorem for block codes where the alphabet is the vertex tive association scheme. This. particular. set of. some. commuta‐. theorem. generalizes the Assmus‐Mattson type theorems mentioned above as well as the original. In proving our results, we invoke several techniques from multivariable polynomial interpolation and from the repre‐ sentation theory of Terwilliger algebras. This is based on a joint work with Hajime Tanaka.. Introduction. 1 We. begin by recalling. Theorem 1.1 n. over. $\Gam a$_{\mathrm{q}. the famous Assmus‐Mattson theorem:. (Assmus. and Mattson. with minimum. weight $\delta$. .. [1, Theorem4.2]).. Let. C^{\perp}. Let C be. a. denote the dual code. linear code. of C. of length. with minimum. Suppose that an integer t (1 \leq t \leq n) exists such that either there are at weights of C^{\perp}in \{1, 2, . . . , n-t\} or there are at most $\delta$^{*}-t weights of C in \{1, 2, . . . , n-t\} Then the supports of the words of any fixed weight in C form a t ‐design (possibly with repeated blocks). weight $\delta$^{*}. .. most $\delta$-t. .. strengthened in many different settings (see [11, 10, 32, 2, 36, 23, 38] for instance). The objective of this paper is to present a theorem which unifies many of the known generalizations and extensions of The theorem mentioned above has been proven and. Theorem 1.1. We advise the reader to check. [29]. for. a more. detailed discussion of the. on Finite Groups, Algebraic Combinatorics and Vertex Operator Algebras Kyoto University, Kyoto, Japan on 6 December 2016 ' Graduate School of Information Sciences, Tohoku University, Sendai, Japan email: moralesjohnvince@ims.is.tohoku.ac.jp *. presented. at the Research. held in RIMS at.

(2) 69. topic. Recent interests in. constructing. combinatorial. designs. from codes. began. when Gul‐. [17] and Harada [18] found new 5‐designs from the lifted Golay code of 24 over \mathb {Z}_{4} (among others) through computer search. Later, these constructions length were further explained and generalized by Bonnecaze, Rains, and Solé [6]. Motivated by these results, Tanabe [35] then obtained an Assmus‐Mattson‐like theorem for linear liver and Harada. symmetrized weight enumerator. Though Tanabe’s Golay code over \mathb {Z}_{4} the technique involves finding the ranks of matrices having quite complicated entries. As a consequence, ver‐ ifying the conditions by manual computations is difficult. Tanabe [37] then presented a simpler version of his theorem, and we can easily check its conditions by hand for the lifted Golay code over \mathbb{Z}_{4}. codes. over. theorem. By. \mathb {Z}_{4} with respect. can. find. to the. from the lifted. 5‐designs. ,. Assmus‐Mattson‐type theorem, we mean a theorem which enables by just looking at some kind of weight enumerator. an. us. to find. of. a. combinatorial t ‐designs. (and. some. natorial. other information such. obtained from such. designs. estimate for the. We shall. see. as. integer. linearity).. as a. theorem does not. t but the wide range of. code. It should be noted that the combi‐. applicability. necessarily give. the best. is commendable.. corresponds a weight enumerator for every s ‐class transla‐ particular, the Hamming weight enumerator is related to. that there. tion association scheme. In. Hamming association schemes which are extensions of 1‐claồs association schemes. Hamming schemes belong to a family called metric and cometric association schemes, and Tanaka [38] showed that Theorem 1.1 can be interpreted and generalized from this point of view. On the other hand, the situation becomes more complicated when s>1 as we are considering an extension of a finer translation association scheme. the. In this paper, we present an Assmus‐Mattson‐type theorem for codes over the vertex some s ‐class translation scheme. In general, the weights of a codeword take the. set of. form. ($\alpha$_{1}, $\alpha$_{2}, \ldots , $\alpha$_{s}). a nonnegative integer and \mathrm{a}_{1}+\cdots+$\alpha$_{n}\leq n. 1 as in Theorem 1.1, weights in a given interval when s but in case s>1 then we speak of the minimal degree of subspaces of the polynomial ring \mathbb{R}[$\xi$_{1}, $\xi$_{2}, $\xi$_{s}] which allow unique Lagrange interpolation with respect to those weights (which are lattice points in \mathbb{R}^{S} ) contained in a given region. When specialized to the case of \mathb {Z}_{4} ‐linear codes with the symmetrized weight enumerator as in [35, 37], the association scheme on the alphabet \mathb {Z}_{4} has two classes R_{1} and R_{2} together with the identity class R_{0} defined by $\alpha$=. where each $\alpha$_{i} is. We count the number of. =. ,. ,. (x, y)\in R_{\dot{ $\eta$}} for i\in. \{0 1, 2 \} ,. ,. \Leftrightarrow. y-x=\pm i (mod4). (x, y\in \mathbb{Z}_{4}). and the extension of this 2‐class association scheme is called the Lee. \mathb {Z}_{4} Our results give a slight extension of Tanabe’s theorem in Assmus‐Mattson‐type theorem for \mathb {Z}_{4} ‐linear codes with the Ham‐ ming weight enumerator due to Shin, Kumar, and Helleseth [31] can also be recovered. In proving our results, we make heavy use of the representation theory of the Ter‐ williger algebra [41, 42, 43], which is a non‐commutative semisimple matrix \mathb {C} ‐algebra association scheme. [37]. Moreover,. the. with respect to. a. over. .. fixed vertex of. an. association scheme..

(3) 70. Preliminaries. 2. In this section. briefly. we. tion schemes and related. thorough. review. some. algebras.. basic concepts. concerning. commutative associa‐. [4, 9, 25]. We advise the reader to check. for. a more. discussion of the topic.. Let X be. a. nonempty finite. Let V denote the vector space. set.. vectors with coordinates indexed. X. by. .. over. \mathb {C} of column. Define the vector \hat{x}\in V for each x\in X such. that y ‐coordinate of \hat{x} is $\delta$_{xy} for all y \in X where $\delta$ is the Kronecker delta function. The set \{\hat{x}| x\in X\} forms an orthonormal basis for V with respect to the Hermitian inner. product \{u, v\rangle =u^{\mathrm{t} \overline{v}. matrices. by. V let. \mathb {C} with. over. left. multiplication.. A_{i}\in \mathrm{M}\mathrm{a}\mathrm{t}_{X}(\mathbb{C}). called. a. for all u, v\in V. Let. ,. the. \mathcal{R}=\{R_{0}, R_{1}, . . . , R_{s}\}. ,. the all. (iii) \{A_{0}, A_{1}, . . . , A_{s}\}. ones. .. denote the \mathb {C} ‐algebra of all. Observe that. denote. \mathrm{M}\mathrm{a}\mathrm{t}_{X}(\mathb {C}). acts. on. X\times X and. partition R_{\dot{ $\eta$}}\subseteq X\times X The pair (X, \mathcal{R}) on. a. .. is. classes if. on s. matrix;. is closed under. (iv) A_{i}A_{j}=A_{j}A_{i}\in M :=\displaystyle \sum_{k=0}^{s}\mathbb{C}A_{k} we. X. identity matrix;. (ii) \displaystyle \sum_{i=0}^{s}A_{i}=J. In this case,. \mathrm{M}\mathrm{a}\mathrm{t}_{X}(\mathrm{C}) by. denote the characteristic matrix of. commutative association scheme. (i) A_{0}=I. Let. .. and columns indexed. rows. conjugate‐tranpose; for 0\leq i,j\leq s.. call X the vertex set and. \mathcal{R}=\{R_{0}, . . . , R_{s}\}. the set of associate classes.. We call V the standard module for the commutative association scheme. (X, \mathcal{R}). .. The Bose‐Mesner algebra M is the commutative subalgebra of \mathrm{M}\mathrm{a}\mathrm{t}_{X}(\mathb {C}) with basis A_{s} There exists a second basis for M consisting of the associate matrices A_{0}, A_{1} E_{s} such that E_{0}+E_{1}+\cdots+E_{s}=I consisting of the matrices E_{0}=|X|^{-1}J, E_{1} and E_{i}E_{j}=$\delta$_{ij}E_{i} for all 0\leq i,j\leq s The matrices E_{0}, E_{1} E_{8} are called primitive We define the matrices P and idempotents. Q by change‐of‐basis .. ,. ,. .. .. .. .. .. ,. .. ,. ,. .. A_{i}=\displaystyle\sum_{j=0}^{s}P_{ji}E_{j} E_{i}=|X|^{-1}\displaystyle \sum_{j=0}^{s}Q_{ji}A_{j}. .. .. .. ,. (1). ,. we. refer to P and Fix x\in X. .. Q. as. the. first. and second. (2). .. eigenmatrix of (X, \mathcal{R}) respectively. ,. We recall the dual Bose‐Mesner. algebra M^{*}(x). \mathrm{M}\mathrm{a}\mathrm{t}_{X}(\mathb {C}). with respect to. for each. x. Define. .. E_{i}^{*}=E_{i}^{*}(x) A_{i}^{*}=A_{i}^{*}(x) (y, y) ‐entries are given by (E_{i}^{*})_{yy} (A_{i})_{xy} and (A_{i}^{*})_{yy}= |X|(E_{i})_{xy} for each y \in X Then M^{*}(x) is the commutative subalgebra of \mathrm{M}\mathrm{a}\mathrm{t}_{X}(\mathb {C}) that has two special bases, \{E_{0}^{*}, E_{1}^{*}, . . . , E_{s}^{*}\} and \{A_{0}^{*}, A_{1}^{*}, . . . , A_{s}^{*}\} The matrices E_{0}^{*}, E_{1}^{*} E_{s}^{*} are called dual primitive idempotents with respect to x and the matrices A_{0}^{*}, A_{1}^{*} A_{s}^{*} are diagonal. matrices. and. in. such that the. integer 0\leq i\leq s. =. .. .. called dual associate matrices with respect to Let We call. ,. .. .. .. ,. ,. .. .. .. ,. x.. T(x) denote the subalgebra of \mathrm{M}\mathrm{a}\mathrm{t}_{X}(\mathb {C}) that is generated by M and M^{*}(x) T(x) the Terwilliger algebra of (X, R) with respect to x Note that T(x) is .. ..

(4) 71. finite‐dimensional and is. T(x). is closed under the conjugate transpose non‐isomorphic irreducible modules for T(x) irreducible T(x) ‐module W\subseteq V define the sets. semisimple. since. map. On the standard module V , any two. orthogonal.. are. For every. ,. W_{s}=\{0\leq i\leq s|E_{i}^{*}W\neq 0\} We call. (resp. exists. W_{s}^{*}=\{0\leq i\leq s|E_{i}W\neq 0\}.. the support and dual support of W , respectively. We say W is thin thin) if \dim(E_{i}^{*}W) \leq 1 for all i (resp. \dim(E_{j}W) \leq 1 for all j ). There. W_{s}, W_{s}^{*}. dual a. and. unique irreducible module for T(x) that is both thin and dual thin for which are both equal to \{0, 1, . . . , s\} We call this the. the support and the dual support primary module for T(x). .. .. We end this section with. a. connection between commutative association schemes. and codes. The reader may refer to [12, 13] for more details. For the rest ofthis section, let C denote a subset of X and let \hat{C} denote the characteristic vector of C\subseteq X In .. this paper, we shall call C a code if 1< |C| < |X| Assume for a moment that C is a code. The inner distribution of C is the vector a= ( a_{0}, al, \in \mathbb{R}^{s+1} where the , a_{s} ) scalars a_{i} are defined by .. .. .. .. a_{i}=|C|^{-1}\langle\hat{C}, A_{i}\hat{C}\rangle=|C|^{-1}\cdot|R_{\dot{ $\eta$}}\cap(C\times C Clearly,. the scalars a_{i}. nonnegative. Observe that using (2),. are. we. obtain. \langle\hat{C}, E_{i}\hat{C}\rangle=|X|^{-1}|C|(aQ)_{i} for every. i(0\leq i\leq s). The vector. aQ. (aQ)_{i}. where. is often referred to. denotes the ith coordinate of the vector the MacWilliams. as. transform. of. aQ\in \mathbb{R}^{s+1}.. a.. Translation Schemes and Extensions. 3. (X, \mathcal{R}). Let. denote. a. commutative association scheme with. s. classes. Assume further. abelian group (written additively) with identity 0 We translation association scheme if for every 0\leq i\leq s and for every z\in X. that X has the structure of. an. .. (X, \mathcal{R}) (x, y)\in R_{\dot{ $\eta$}} implies (x+z, y+z)\in R_{ $\eta$} For the rest of the section, assume that (X, \mathcal{R}) is a translation association scheme and we shall pick the identity element as the. call we. a. have. .. base vertex when. dealing. with. We show that there exists. an. turns out that this dual is also. Terwilliger algebras association scheme. that is dual to. (X, \mathcal{R}). .. It. translation association scheme.. a. Let X^{*} denote the character group of X with we. of translation association schemes.. (X^{*}, \mathcal{R}^{*}) identity. element. L. .. To each $\epsilon$\in X^{*}. associate the vector. \displaystyle \hat{ $\epsilon$}=|X|^{-1/2}\sum_{x\in X}\overline{ $\epsilon$(x)}\hat{x}\in V. By the orthogonal relations of the characters, forms. an. orthonormal basis of V. for the associate matrices. A_{0}, A_{1}. \langle A_{i}\hat{ $\epsilon$},. ,. we. observe that the set. We claim that these basis vectors. .. .. .. .. ,. A_{s} In particular, .. one. routinely. \hat{$\tau$}\rangle=\left\{ begin{ar y}{l 0&\mathrm{i}\mathrm{f} $\epsilon$\ eq$\tau$\ \sum_{x\inX_{i}\overline{\mathcal{E}i(x) &\mathrm{i}\mathrm{f} $\epsilon$= \tau$ \end{ar y}\right.. \{\hat{ $\epsilon$}| $\epsilon$ \in X^{*}\}. are. eigenvectors. obtains.

(5) 72. for every $\epsilon$, $\tau$\in X^{*} where X_{i}=\{x\in X| (0, x)\in R_{ $\eta$}\} for each 0\leq i\leq s Consequently, each vector \hat{$\epsilon$} is contained in one of the subspaces E_{i}V of V Thus, we have a partition .. .. X^{*}=X_{0}^{*}\sqcup x_{1}^{*}\mathrm{u}\cdots \mathrm{u}x_{ $\epsilon$}^{*}, given by X_{i}^{*}=\{ $\epsilon$\in X^{*} : \hat{ $\epsilon$}\in E_{i}V\} (0\leq i\leq s) Define of nonempty binary relations on X^{*} by. the set. .. \prime \mathcal{R}^{*}=\{R_{\mathrm{c} ^{*}, R_{1}^{*}, . . . , R_{s}^{*}\}. R_{ $\eta$}^{*}=\{( $\epsilon$, $\eta$)\in X^{*}\times X^{*} : $\eta \epsilon$^{-1}\in X_{i}^{*}\} (0\leq i\leq s) pair (X^{*}, \mathcal{R}^{*}) (X, \mathcal{R}) Suppose that P^{*}. It turns out that the the dual of. .. (X^{*}, R^{*}) respectively.. Then. ,. .. again a translation association scheme called Q^{*} are the first and second eigenmatrices of P^{*}=Q and Q^{*}=P and that is. and. ,. P_{ji}=\displaystyle \sum_{x\in X_{i} \overline{ $\epsilon$(x)} ( $\epsilon$\in X_{j}^{*}) , Q_{ji}=\sum_{ $\epsilon$\in X_{i}^{*} $\epsilon$(x) (x\in X_{j}) We view V. We call. together a. with the basis. code C in X. moment that C is. an. an. \{\hat{ $\epsilon$}: $\epsilon$\in X^{*}\}. as. additive code if C is. additive. code,. inner distribution. We claim that. the standard module for. a. subgroup of. and let the vector. a_{i}=|X_{i}\cap C|. .. a=. X. ( a_{0}, al,. for each 0\leq i\leq s. .. .. (X^{*}, \mathcal{R}^{*}). .. Assume for the. .. .. .. ,. a_{s} ) denote its. This follows from. the fact that every element z\in X_{i}\cap C can be expressed as y-x where x, y\in C in exactly |C| ways. In this case, the vector a is also called the weight distribution of C.. The duat code of C is the. subgroup C^{\perp}. C^{\perp}= { $\epsilon$\in X^{*} It turns out that. weight. |X_{i}^{*}\cap C^{\perp}|. distribution of. The group operation as a code in X by. on. X^{*} is. fixing. $\epsilon$(x)=1. by. for all x\in C }.. |C|^{-1}(aQ)_{i} (0 \leq i \leq s). =. C^{\perp}.. in X^{*}. :. in X^{*} defined. \mathrm{a}. multiplicative.. that. so. In many. |C|^{-1}(aQ) gives. cases we. (non‐canonical) isomorphism. a. code. (x\mapsto$\epsilon$_{x}). such. may view. X\rightarrow X^{*}. the. that. $\epsilon$_{x}(y)=$\epsilon$_{y}(x) (x, y\in X) Thus, For. the dual code of. more. an. additive code in X becomes. details about translation association. Chapter 6], [9, §2.10],. and. (3). .. again. schemes,. an. additive code in X.. the reader may refer to. [12,. [25, §6].. For the rest of this paper, we will fix an integer n\geq 2 Delsarte [12, §2.5] gave a a new commutative association scheme from (X, \mathcal{R}) with vertex set X^{n} .. construction of as. follows. For. vertices. x=. a. sequence. $\alpha$=. ($\alpha$_{1}, $\alpha$_{2}, \ldots , $\alpha$_{s}). \in \mathbb{N}^{s} let ,. (x_{1}, x2, . . . , x_{n}) y= (y_{1}, y2, . . . , y_{n}) c(x, y)=(c_{1}, c_{2}, \ldots , c_{s})\in \mathrm{N}^{S} where ,. | $\alpha$| =\displaystyle \sum_{i=1}^{s}$\alpha$_{i}. .. For any two. \in X^{n} , define the composition of x, y. to be the vector. c_{i}=|\{l:(x_{\ell}, y_{\ell})\in R_{ $\eta$}\}| (1\leq i\leq s) Let S denote the set of all relation. R_{\mathfrak{v}. on. X^{n}. possible compositions. For. .. every $\alpha$\in S , define the. by. R_{ $\alpha$}=\{(x, y)\in X^{n}\times X^{n} : c(x, y)= $\alpha$\}.. binary.

(6) 73. pair (X^{n}, \{R_{ $\alpha$} : $\alpha$\in S\}) is a commutative association scheme (X, \mathcal{R}) of length n We shall identify its standard module with. Then it follows that the. called the extension of. V^{\otimes n}. .. :=\hat{x}_{1}\otimes\hat{x}_{2}\otimes\cdots\otimes\hat{x}_{n} for every x= (x_{1}, x2, . . . , x_{n}) \in X^{n} For every associate matrix A_{ $\alpha$} is the characteristic matrix of R_{ $\alpha$} \subseteq X^{n} and is then. that \hat{x}. so. $\alpha$\in S the ,. .. given by. A_{ $\alpha$}=\displaystyle \sum_{i_{1},i_{2},\ldots,i_{n} A_{i_{1} \otimes A_{i_{2} \otimes\cdots\otimes A_{i_{n} where the. sum. is. i_{1}, i_{2}. over. ,. .. .. .,. (4). ,. i_{n}\in \mathrm{N} such that. \{i_{1}, i_{2}, . . . , i_{n}\}=\{0^{n-| $\alpha$|}, 1^{$\alpha$_{1}} , 2^{ $\alpha$ 2}, . . . , s^{$\alpha$_{s}}\} particular, the Bose‐Mesner algebra M of (X^{n}, \{R_{ $\alpha$} : $\alpha$\in S\}) coincides n^{\mathrm{t}\mathrm{h} symmetric tensor space of M Similar expressions hold for the primitive idempotents, dual idempotents, and the dual associate matrices of (X^{n}, \{R_{ $\alpha$} : $\alpha$\in S denoted by the E_{ $\alpha$}, E_{ $\alpha$}^{*} and the A_{ $\alpha$}^{*} respectively. Recall that if x_{0} is the identity element in X then the vertex x_{0} := (x0, x0, . . . , x_{0}) is the identity element in X^{n} and is then chosen as the base vertex. We denote the corresponding dual Bose‐Mesner algebra and the Terwilliger algebra by M^{*} and T respectively. We also consider the as. multisets. In. with the. .. ,. ,. ,. partition. X^{n}=\sqcup(X^{n})_{ $\alpha$} $\alpha$\in S (X^{n})_{ $\alpha$}=\{x\in X^{n} : (x_{0}, x)\in R_{ $\alpha$}\}.. where Let. \{e_{i} : 1\leq i\leq s\}. where $\alpha$_{0} trices of. [22]. .. It. can. be. checked that. routinely. A_{\mathrm{e}_i =\displaystle\sum_{$\alpha$\in mathrm{N}^s ,|$\alpha$|\leqn}(\sum_{j=0}^{s$\alpha$_{j}P_{ji})E_{$\alpha$},A_{e i}^{*=\sum_{$\alpha$\in mathrm{N}^s ,|$\alpha$|\leqn}(\sum_{j=0}^{s$\alpha$_{j}Q_{ji})E_{$\alpha$}^{*. (5). ,. :=n-| $\alpha$| More generally, Mizukawa and Tanaka [27] described the eigenma‐ (X^{n}, \{R_{ $\alpha$} : $\alpha$\in S\}) in terms of certain s ‐variable hypergeometric orthogonal .. polynomials to. be the standard basis of \mathbb{R}^{S}. and. that. generalize. the Krawtchouk. polynomials.. The reader may also refer. [21].. Suppose (X, \mathcal{R}) is a translation association scheme and let (X^{n}, \{R_{ $\alpha$} : $\alpha$\in S\}) is length n of the association scheme (X, R) To this end, we define a corresponding weight enumerator of C for every additive code C in X^{n} and we recall the extension of. a. .. generalization of the so that $\xi$_{0}, $\xi$_{1}. sequence. well‐known MacWilliams ,. .. .. .. ,. $\xi$_{s}. are. identity.. Let. $\xi$=($\xi$_{0}, $\xi$_{1}, \ldots, $\xi$_{s}). indeterminates. For every element $\alpha$\in S ,. we. be. a. define. $\xi$^{ $\alpha$}=$\xi$_{0}^{n-| $\alpha$|}$\xi$_{1}^{$\alpha$_{1} $\xi$_{2}^{$\alpha$_{2} \ldots$\xi$_{s}^{$\alpha$_{8} . Now,. we. consider. Define the. a. code C in X^{n} with. polynomial w_{C}( $\xi$). in. corresponding. inner distribution. a=(a_{ $\alpha$})_{ $\alpha$\in S}.. \mathbb{R}[ $\xi$]=\mathbb{R}[$\xi$_{0}, $\xi$_{1}, . . . , $\xi$_{s}] given by. w_{C}($\xi$)=\displaystyle\sum_{$\alpha$\inS}a_{$\alpha$} \xi$^{$\alpha$}. Since. (X^{n}, \{R_{ $\alpha$} : $\alpha$\in S\}). weight. distribution of C. .. is. a. scheme, the vector a is also the polynomial wc( $\xi$) as the weight enumerator. translation association. We refer to the.

(7) 74. of C. It should be remarked that. .. dual to each other. It. (X^{n}, \{R_{ $\alpha$} : $\alpha$\in S\}). and. (X^{*n}, \{R_{ $\alpha$}^{*} : $\alpha$\in S^{*}\}). are. be shown that. can. w_{C^{\perp}}( $\xi$)=|C|^{-1}w_{C}( $\xi$ Q^{\mathrm{T}}). .. This. generalizes the well‐known MacWilliams identity. This equation implies that weight enumerator of the dual code C^{\perp} can be easily obtained from the weight. the. enumerator of the code C.. Codes. 4. \mathb {Z}_{4}. over. In this section, we consider abelian group X=\mathbb{Z}_{4} as the vertex set and discuss different examples of translation association schemes (X, \mathcal{R}) We also construct the extension (X^{n}, \{R_{ $\alpha$} : $\alpha$ \in S\}) of length n of the translation scheme (X, \mathcal{R}) and describe the corresponding weight enumerator of an additive code C in X^{n}. .. For our first example, we consider the partition \mathcal{R}= \{R_{0}, (X\times X)\backslash R_{0}\} so that (X^{n}, \{R_{ $\alpha$} : $\alpha$\in S\}) is an extension of a one‐class association scheme. The association scheme (X^{n}, \{R_{ $\alpha$} : $\alpha$\in S\}) is called the Hamming association scheme. The Hamming. schemes. are. they belong to a family of metric and corresponding weight enumerator of any additive Hamming weight enumerator.. well‐studied association schemes and. cometric schemes. In this case, the. code in X^{n} is called the. our second example, we consider the partition \mathcal{R}=\{R_{0}, R_{1}, R_{2}\} of X\times X such R_{\dot{ $\eta$} \{(x, y) \in X\times X | y-x=\pm i (\mathrm{m}\mathrm{o}\mathrm{d} 4)\} for each 0\leq i \leq 2 Observe that the pair (X, \mathcal{R}) is a two‐class translation association scheme. We refer to the extension (X^{n}, \{R_{ $\alpha$} : $\alpha$\in S\}) as a Lee association scheme over \mathb {Z}_{4} The Terwilliger algebras of Lee association schemes over \mathb {Z}_{4} are described in [28]. Now let C denote an additive. For. that. =. .. .. code in X^{n} and let to. wc( $\xi$). as. Finally,. the we. w_{C}( $\xi$). denote the. symmetrized weight. consider the. corresponding weight. enumerator of C. .. We refer. enumerator of C.. partition \mathcal{R}=\{R_{0}, R_{1}, R_{2}, R3\} of. X\times X. given by. R_{1}=\{(0,1) (1, 2), (2, 3), (3, 0 R_{2}=\{(0,2) (1, 3), ( 2, 0) (3, 1 R_{3}=\{(0,3) (1, 0) (2, 1), (3, 2) \} ,. ,. ,. so. that. (X^{n}, \{R_{ $\alpha$} : $\alpha$\in S\}). is. case, the. an. corresponding weight complete weight enumerator.. 5. ,. ,. extension of. a. three‐class association scheme. In this. enumerator of any additive code in X^{n} is called the. Main Results. section, we present the main theorem (Theorem 5.3) as well as some supple‐ improve the main theorem (see [29] for the proofs). To do this, we need to recall some concepts from polynomial interpolation (see [14] for more details). In this. ments that.

(8) 75. Let F be a finite set of points in \mathbb{R}^{S} We call a linear subspace \mathscr{L} of \mathbb{R}[$\xi$_{1}, $\xi$_{2}, \cdots, $\xi$_{s}] interpolation space with respect to F if for every f\in \mathbb{R}[$\xi$_{1}, $\xi$_{2}, . . . , $\xi$_{s}] there exists a unique g\in \mathscr{L} such that f(z) =g(z) for every point z\in F If, in addition, this g satisfies then we refer to \mat h scr { L } as a minimal always \deg f \geq \deg g degree interpolation space. Let \mathscr{M}(F) denote a minimal degree interpolation space with respect to F and .. an. ,. .. ,. define. $\mu$(F)=\displaystyle \max\{\deg f : f\in \mathscr{M}(F)\}. construction of a minimal degree interpolation space \mathscr{M}(F) due to de Boor For every non‐zero element f (see [7, 8 \displaystyle \sum_{i=0}^{\infty}f_{i} in the ring of formal series \mathbb{R}[ $\xi$_{1}, $\xi$_{2}, \cdots, $\xi$_{s}] where f_{i} is homogeneous of degree i define. We recall. a. and Ron power. =. ,. f\downarrow=f_{i_{0}}, where us a. i_{0}=\displaystyle \min\{i:f_{i}\neq 0\} Conventionally, .. construction for. a. minimal. we set 0\downarrow:=0 The theorem degree interpolation space \mathscr{M}(F) .. ([7, 8 Let F be a finite set of points \mathbb{R}[ $\xi$_{1}, $\xi$_{2}, . . . , $\xi$_{s}] spanned by the exponential functions Theorem 5.1. in \mathbb{R}^{S}. Let á be the. .. \displaystyle \exp(\sum_{i=1}^{s}z_{i}$\xi$_{i}) ( z_{1}, z2, . . , z_{s})\in F) Then the. below. gives. .. subspace of. .. subspace. \displaystyle\sum_{f\in\mathscr{E}\mathb {R}f_{\downar ow}\subset\mathb {R}[$\xi$_{1},$\xi$_{2},. ,$\xi$_{s}] is. a. minimal. Theorem 5.1. degree interpolation immediately. space with. leads to the. respect. to F.. following formula for $\mu$(F). which is well suited. for computer calculations:. Supplement. 5.2. For every. smallest m\in \mathrm{N}. for. which the. finite set F of points polynomials. in \mathbb{R}^{s} , the scalar. $\mu$(F) equals. the. \displaystyle \sum_{k=0}^{m}(\sum_{i=1}^{s}z_{i}$\xi$_{i})^{k} ( z_{1}, z2, \cdots z_{s})\in F) are. linearly independent.. We. see. that. \mathscr{M}(F). well‐defined that. is,. exists but is not it is. independent. unique. However, of the choice of. We retain the notation of Section 3. For every. x=. it. can. \mathscr{M}(F). be shown that. $\mu$(F). is. .. (x_{1}, x2, . . . , x_{n})\in X^{n}. ,. define. \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(x)=\{P:x_{\ell}\neq x_{0}\}\subset\{1, 2, . . . , n\}. We call theorem.. \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(x). the support of. x. (with respect. to x_{0}. ).. We. now. present. our. main.

(9) 76. Theorem 5.3. Let C denote. code in X^{n} and let. a. corresponding weight enumerator. merator of the dual code C^{\perp}of C.. w_{C^{\perp}}( $\xi$). Let. =. Let. wc( $\xi$). =. \displaystyle \sum_{ $\alpha$\in S}a_{ $\alpha$}^{*}$\xi$^{ $\alpha$}. \displaystyle \sum_{ $\alpha$\in S}a_{ $\alpha$}$\xi$^{ $\alpha$}. denote the. F_{r}=\{ $\alpha$\in S : r\leq| $\alpha$| \leq n-r, a_{ $\alpha$}\neq 0\} (1\leq r\leq \lfloor n/2\rfloor). denote its. weight. enu‐. ,. and let. $\delta$^{*}=\displaystyle \min\{| $\alpha$|\neq 0: $\alpha$\in S Suppose. that. an. integer t(1\leq t\leq n). and. a_{ $\alpha$}^{*}\neq 0\}.. is such that. $\mu$(F_{r})<$\delta$^{*}-r (1\leq r\leq t). (6). .. Then the multiset. (7). \{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mathrm{x}) : x\in(X^{n})_{ $\alpha$}\cap C\} is. for every $\alpha$\in \mathrm{N}^{S} with. t ‐design. a. We ments. use. Theorem 5.3. improve the. Supplement. together with the following “supplements” and these supple‐. main theorem.. 5.4. Let C be. code in X “. Assume that. a. (7). K\subset S such that the multiset. (6). | $\alpha$|\leq n.. in Theorem 5.3 may be. \dot{u}. a. t ‐design. for. we are. We call. a. subset C of X^{n}. if,. set. a. Then the condition. replaced by. $\mu$(F_{r}\backslash K)<$\delta$^{*}-r (1\leq $\tau$\leq t) the base vertex x_{0} ) number. in advance. given. every $\alpha$\in K.. a. weakly. t ‐balanced. $\Lambda$\subset\{1, 2, . . . , n\}. for any. arrayl. and. .. over. $\gamma$\in S. (X, \mathcal{R}) (with. respect. | $\gamma$| \leq| $\Lambda$|\leq t. such that. ,. to. the. |\{x\in C:(x_{i})_{i\in $\Lambda$}\in (X^{| $\Lambda$|})_{ $\gamma$}\}| depends only. Supplement C \dot{u}. an. on. | $\Lambda$|. 5.5.. and $\gamma$.. Suppose. additive code in X^{n}. that .. (X, \mathcal{R}). (X^{*n})_{ $\alpha$}\cap C^{\perp}is. that, for. is. Assume that. every $\alpha$\in L, the scalar $\delta$^{*} in Theorem 5.3 may be. a. translation association. we are. in advance. given. weakly t ‐balanced replaced by a. \displaystyle \min\{|\mathrm{a}| : 0\neq $\alpha$\in S\backslash L, a_{ $\alpha$}^{*}\neq 0\}. array. ,. 5.6 below. and hence t ,. Supplement scalars z_{i\ell}. $\sigma$\in \mathrm{G}\mathrm{L}(\mathbb{R}^{S}). 5.6. Let F be. a. finite. (1 \leq i \leq s, \ell \in \mathrm{N}) such that. z_{ik}\neq zu. ,. a. '. as. Then. allows. us. to estimate. result about minimal. set of points in \mathbb{R}^{S} Suppose that there are real positive integer m and a linear automorphism. whenever. $\mu$(F)\leq m.. This term is meant. .. (8). .. ,. k\neq P. ,. and that. $\sigma$(F)\subset\{(z, z, \ldots, z_{s$\alpha$_{8}})\in \mathbb{R}^{S}: $\alpha$\in \mathbb{N}^{s}, | $\alpha$|\leq m\} Then. set L\subset S such. (X^{*}, \mathcal{R}^{*}). over. .. was inspired by [37, Theorem 2], and by geometrical considerations. It is a general degree interpolation spaces.. Supplement. $\mu$(F). scheme, and that a. only provisional; cf. [34].. .. (9).

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