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B. Mounits
NEW UPPER BOUNDS FOR NONBINARY CODES
The paper presents new upper bounds for non-binary codes. The bounds can be obtained by linear and semidefinite programming.
bounds, codes, programming.
Abstract - New upper bounds on codes are presented. The bounds are obtained by linear and semidefinite programming.
INTRODUCTION
One of the central problems in coding theory is to find upper bounds on maximum size Aq(n, d) of a code of word length n and minimum Hamming distance at least d over the alphabet Q of q > 2 letters. Let us provide Q with the structure of an Abelian group, in an arbitrary way.
In 1973 Delsarte proposed a linear programming approach for bounding the size of cliques in an association scheme. This bound is based on diagonalizing the BoseMesner algebra of the scheme. To obtain bounds on Aq(n, d), Delsarte introduced the Hamming scheme H(n, q), which is generated by action of a group of permutations of Qn that preserve the Hamming distance.
In 2005 Schrijver gave a new upper bound on A2(n,d) using semidefinite pro-
3
gramming, which is obtained by block-diagonalizing the (~^) - dimensional
Terwilliger algebra of H(n,2). The semidefinite programming bound for Aq(n,d), based
on the block-diagonalizing the (----) - dimensional Terwilliger algebra of H(n, q) was
presented later by Gijswijt, Schrijver and Tanaka.
In this work we introduce an association scheme which is generated by a subgroup of permutations of Qn that preserve not only the Hamming distance, but also the "type" of the difference of vectors. The dimension of the Bose-Mesner algebra of this
n+Q—1 1
scheme is ( ). We also describe the ( t ) - dimensional Terwilliger algebra
of this new scheme. In particular, we have found that the orbits of Qn x Qn x Qn under the action of the subgroup are characterized by certain q x q matrices.
With these two algebras in hand, we derive a linear programming bound and a semidefinite programming bound for Aq(n, d) which generalize the bounds above. For the binary case, our scheme and the Hamming scheme H(n, 2) coincide.
ASSOCIATION SCHEMES AND THE LP BOUND
Let G -{gi = 0, g2, ..., g|G} denote an (additively written) arbitrary finite abelian group with zero element 0, and Gn = G x G x... x G denote an abelian group with respect to componentwise sum. For an integer n we denote
N n- = {(^gl, ■■■, ^glGl )■ ttg G t0,1, ... , ft}, ^¡gsG ^}.
Define a function y: Gn^N°n as follows:
W(x) -= (cgi{x),cg2{x),..., Cg\G\(x)), Cg(x)=\{i:xi = g}\.
A nonempty subset C of Gn is called a code of length n. For a set S Q Nfi we
define
AG(n, S) := max{|C| - C Q Gn, y (y - x) GS Vx, y G C} .
For aG Nfi let Ra be a relation Ra := {(x, y) G Gn x Gn : y (y - x) = a} and denote R = {Ra} a G N°n.
Let H denote a group consisting of the permutations of Gn obtained by permuting the n coordinates followed by adding a word from Gn, i.e.,
H = {n (•) + u: n G Sn, u G Gn}.
It is obvious that H acts transitively on Gn. H has a natural action on Gn x Gn
given by h(x,y) := (hx, hy). The following lemma states that the orbitals {(hx, hy) :
hG H} form the relations of R.
Lemma 2.1. For any a G N°n and x,y G Gn such that (x,y) G Ra there holds Ra = = {(hx,hy) : h G H}.
Proof: Let x, y G Xbe such that y (y - x) = a. Thus, for h = n (•) + v G H, hy - hx = (ny + u) - (nx + u) = n(y - x) and
y (hy - hx) = y (y - x) (1)
which implies that {(hx, hy) : h G H} QRa
On the other hand, we have to show that for any (x,y) G Ra there exists h G H such that (x, y) = (hx, hy). One can see that exists h0 G H such that hex = 0 and h0y -
ua, where a0 agi aglGl
ua = 0—0... QiQi ...gi... gid ■■■g|G|,
namely he(•) = tcoW - nx for some n0 G Sn- Similarly, there exists h1 G H such that h1x = = 0 and h1 y - ua, namely h1(^)=n1(^) - n1 x for some n1 G Sn. Thus,
h(^) = h1-1 he(•) = n1-1 n0(^) + x - nf1 n0(x)
satisfies (hx,hy) = (x,y) which proves the required inclusion.
Theorem 2.2. (Gn,R) is a commutative association scheme with (n*|C| *) rela-
' 7 V |G|-1 J
tions.
Proof: It is well known (see for example [1]) that the orbitals from a group action form relations of an association scheme. For (x, y) G RY denote
Z(x,y): = {zG Gn : (x,z) G Ra (z,y) G Rp},
7.(x,y) = {zGGn : (x,z) G Rp, (z,y) G Ra}.
Since z G Z(X,y) ^(-z) G 7. (-y,-x) we conclude that
pYa,P = |Z(x,y)| = 1 Z (-y,-x)\= pYP, a .
Note that the number of relations is equal to the number of (|G| - 1) - tuples of nonnegative integers (ag2, ., ag|G) such that ag2 + ... + ag|G| < n.
Let Da denote the adjacency matrix of the relation Ra, i.e.,
(D ) = iW (x,y)GRa,
a x,y I 0, otherwise .
The matrices {Da} a G N% form a basis of a commutative - dimen-
sional Bose-Mesner algebra AGn of the scheme (Gn, R).
In general, (Gn, R) is a non-symmetric association scheme.
For a G NGn, the inverse R a-1 = {(y, x) : (x, y) G Ra} of the relation Ra is given by R a"1 = Rg where
a := (a52,-,a5|C|) , = a_g.. (2)
It's easy to see that the valency of the relation Ra (and of Rq) is ua = p^a0’ "’0) =
= ( " ).
\a0’ag2’—’ag\G\/
Consider the association scheme (Gn,R), where R={Ra}, Ra = Ra U R^1. This is symmetric association scheme. Note that
Da = Da + Dq
are symmetric matrices. We denote by AGn the Bose-Mesner algebra of {Gn, R) and by Gn = [Xu}uGGn the group of characters. The next theorem gives more details about the symmetric scheme.
Theorem 2.3. The unitary matrix U which diagonalizes the AGn is given by
W*,y=^iÆ(y) .
The primitive idempotent J a, a e N%, is the matrix with (x, y) entry
(Ja)x,y rin S zeGn ^y-x (^) ■
1 1 q>(z)e{a,a}
The eigenvalues are given by
Pp(a) = Qp(a) = X zecn Xu(z)
(4)
ip(z)e{a,a}
where u G Gn is any word with y(u) G {a,a}. For a = (a0, ., ag|G|)G Nfi there holds
ygeG* J p-(p0,...,p ^
Qp (a).
(Po ,.,P£(|G|)eWG Pg2 + -+pg\G\=k
Where Kk (x) is the Krawtchouk polynomial of degree k.
A. Association Scheme for G = Z3.
Let us look at an example for G = Z3 = {0, 1, 2}. For convenience we will omit a0.
N% = {a - (ai, a2) : ai + a2 < n}.
Thus, the number of relations in a non-symmetric scheme (Z”, R) is |R| = = (”2 2), and the number of relations in the symmetric scheme (Z™,R is
p+S<P1
q+t<p2
x e2ni/3(p+q+s+t)(e2ni/3(q+s) + (^ — ^ )g2rei/3(p+t))
We list here few polynomials:
<?(1,0)((«1,«2)) = 2n- 3(at + a2),
Q(i,i)((«i,«2)) = 2(n-“21-“2) -
(at + a2)(n - («i + a2)) - «i«2 + 2 Q1) + 2 Q2),
<?(2,o)((«i,«2)) = 2(n-“21-“2) -
Oi + a2)(n - («! + a2)) + 2ai«2 - Q1) - Q*),
2 ¿/ at = a2(mod 3),
<?(n,o)((«i,«2)) = {2 1‘ -1
else.
B. The Linear Programming Bound.
For a code CGGn let (aY)YG Nfi denote the inner distribution of C, i.e.,
\jRy nCx C\
ay= \c\ .
Clearly, we have
a(n,0,...,0) = 1, ^yGN^aY = \^\.
The Delsarte's linear programming bound is given in the following theorem. Theorem 2.4. (LP bound) For any positive integer n and set SQ Nfi such that (n, 0, ... , 0) GS
AG(n,S) < [max
YeN%
subject to the constraints
a(n,0,... ,0) = 1
aY = 0 for y&S,
^ Qa(Y)aY > 0,a G Nfi.
YGN%
Where Qa(y) is given in (4).
THE TERWILLIGER ALGEBRA OF (Gn, R)
We will now consider the action of H on, ordered triples of words, leading to noncommutative algebra TGn containing the Bose-Mesner algebra. Let M,{G) be the following set of matrices:
Mn(G) : = AeC\G\x\G\■ (A)gijgj G {0,1, ...,n} and Zgi,gj(A)gi,gj = n}.
For any matrix A G Mn(G) we define three vectors r(A), c(A), p(A) G N„ by
"(^) ( ^ (^)o’5, " , ^ (^)g\G\,g j,
\gGG gGG J
:G4) = (I (^)0,g, ■■■ , (A')g,g\G\ J,
\gGG gGG J
P(A) {X‘gGG(^')g,g,Y1gGG(^L)g(£+g2), ■ ,Y1gGG(^L)g,(g+g\G\)]. (5)
To each ordered triple (x, y, z) G Gn xGn x Gn we associate the matrix
Tp(x,y,z): = A*,z G Mn(G)
where
(Ay,z)gl,g] = |{^: O - x)k = 9i, (z - x)k = 9j]\ ■
Note that y(y - x), y(z - x) and y(z - y) are uniquely determined by the Ay,z:
y(y - x) = r(A*z), y(z - x) = c (A*,z), y(z -y) =p (AxViZ). (6)
If we define
Xa : = {x, y, z} G Gnx Gn:Tp(x, y, z) = A}
for A GMn(G), we have the following.
Lemma 3.1. The sets XA, A G Mn(G), are the orbits of Gn x Gn x Gn under the action of H.
Proof: Let x, y, z G Gn and let ip (x, y, z) = A. For h = n(^) + v G H we have from (1) ($(hx, hy, hz))gi,gj = |{k: (hy - hx)k = gi, (hz - hx)k = gj}| = |{k: (n(y - x))k = gi, (n(z - x))k = gj}|=|{k: (y - x)k = gi, (z -x)k = gj}| = (A)p,g j= (i/J(x,y,z))gi,gj, which implies
for any heH.
Let A eMn(G). To show that H acts transitively on XA it suffices to show that for every (x, y, z) eXA there is heH such that (hx, hy, hz) only depends on A. For convenience, we denote y(y - x) = a and y (z - x) = p. Let n0eSn be such that
a0 agi aglGl
no(y- x)=«a=0..........0- 'gigi-gi-'giGi -s,\g\‘.
Now, let n1eSn be such that
n1ua and n1n0(z - x) = up,
where
a a0 0....0 a9i a3\G\
gigt... gi 3iGi...gici
P o...oo)ffi>o...ffH-ffl GI(A)i,glGI
Thus,
h = TC1TC0M - nino(x).
Denote the stabilizer of 0 e Gn in H by H0. For A eMn (G), let MA be the |G|n x |G|n matrix defined by:
(M ) : = (M/ 4>(0,y,z) = A,
A y,z : 1 0, otherwise.
Note that
Ml = MAr.
Let zGn be the set of matrices
^ xAMa,
AeMn(G)
where xA e C. From the Lemma 3.1 it follows that zGn is the set of matrices that are stable under permutations aeH0of the rows and columns, i.e., fo any aeH0 and MA,
(MAy,z = (MA)oy, az .
Hence TGn is a complex matrix algebra called the centralizer algebra of H0.
Since
MaMb = 0 if c(A)± r{B) .
it follows that zGn is a noncommutative algebra. The MA constitute a basis for zGn, and hence
dimr,
Note that the algebra zGn contains the Bose-Mesner algebra AGn; for ye Nfi we have (recall (5))
Dy= ^ MA,
AeM n(G)
p(a)=y
Let t denote the Terwilliger algebra of the association scheme (Gn, R) (with respect to 0). It is the complex matrix algebra generated by the adjacency matrices of the scheme {D7}7e Nfi and the diagonal matrices {£y}Ye Nfi defined by
■ = j1 Ô'
if (0, x) e RY othewise
Theorem 3.2. The algebras rGn and t coincide.
Proof: We have already seen in (7) that zGn contains the adjacency matrices DY. Note that
= MAy,
where Ay= diag(y0, yg2, ..., yg|G|) eMn{G). Hence t is a subalgebra of zGn. Now we show the reverse inclusion. For ye N% with ygi > k and gj e G, gi * gj, define
y(k,g„gj) e N% by f Ygi if I * i,j,
(y(k,gi,g]))gi =j Ygi - k if l = i,
[yg. + k if l=j.
Also define the zero-one matrices:
NY(k,gi,gj) EyD(n-k, 0, 0,k,0,....,0) Ey(k,gi,gj)
where at the index of the matrix D, k appears in the (gj - gi) coordinate. Observe that
(NY(k,gi,gj))y,z = 1^(0, y, z) eXA,
Where
rYgl if i = m and i ^ i,
Yai-k if l = m = i,
k if (l,m) = (i,j),
0 otherwise.
I. Semidefinite Programming Bound
For h e H denote the characteristic vector of h(C) by X(hC) (taken as a column vector). For a word x e Gn, let hx e H be any automorphism with hx(x) = 0, and define
Rx = 1 ^ jo'(Mc')) (Xa(hx(c)))T.
\Hn \ ¿—l
oeHn
Next define the matrices R and R' by
R : = Sxec Rx,
R ' (\G\n-\C\ ^xeGn\CRx.
As the Rx, and hence also R and R', are convex combinations of positive semidefinite matrices, they are positive semidefinite. By construction, the matrices Rx, and hence the matrices R and R' are invariant under permutations a eH0 of rows and columns and hence they are elements of the algebra To*. Define the numbers
2a := | (C xC xC)nXAl
and let
jua : = |({0}xG"xG")nXA| be the number of nonzero entries of Ma. It is easy to see that
rC^)o< r(A■')g2, — , rC^)a|G|/ l^ec V^^i,0,(■A)gl,g2, ■" ' C^Si^lGl
R
AeMn(G)
R' =
|C|
|c|n- |C|
AeMn(G)
where
*A = (I^U) %.
Proof: Denote by (A,B) := tr(A*B), the standard inner product on the space of complex |Gn| x |G"| matrices. Observe that the matrices MA are pairwise orthogonal and that (MA,MA) = ^A for A G Mn (G). Hence
(r,ma) = = ||i^I(M xCx C)nX,|=i^A,
xec xec
implies that
^ (R)y,z = À.A
(0,y,z)eXA
which is the total number of l's (with repetitions) in positions where R and MA are both nonzero. From the symmetry, each entry in MA is counted the same number of times which is (^A)-1^A. Thus the first claim follows:
R = T,AeMn(G)~(R, MA) = TlAeMn(G)XA^A.
Now, the matrix
T:= № + (|cr-№'= ^ Rx =
xeGn
= ^LV V x<r(_hx(C))(x<j(_hx(C)))T =
|^0| ¿-1 ¿-1
xeGn aeHa
V xa(c)(xa(c))T = — Y xh(c)(xh(c))1
^ ^ |H0 | ^
xeGn aeHn heH
is invariant under permutation of the rows and columns by permutations h e H and hence is an element of the Bose-Mesner algebra say
T = Zyew£ ^y^y.
Note that for any z e Gn with y(z) = y, we have
by = (T)0,z = |C|(R)0,z + (|G|n - |C|)(R')0,z .
From the definition of R'
R (\G\n- \C\) Rx
Vl 1 xeGn\C
follows that for x e Gn \ C, holds 0 ^ hx(C) and 0 ^ a(hx(C)) for any a e H0. Therefore, (R')0,z = 0 and we obtain
bY = (T)0,z = |C|(R)0,z = |C| Y1AeMn(G)XA(MA)0,z = K^y^
where (y) denotes a matrix whose first row is a vector y and the rest of the rows are zero vectors. Hence we have
(|G|n - |C|)R' = T - |C| R = |C| Y.AeMn(R)(xfoA)) - xa)Ma.
Finally, note that
«(PM»)'1'1(,>(.«) = \c\^r,At 'c x c n R|,U)\ = Cm
X{v(A)) = (\C\m(p(^))) \p(A)) = \rL \c X C n Rp(A) \ =
= (\C\mAp(^) = X^p(A).
Since for yeWjf,
*—
cec e'ec
ip(c'-c)=Y
It follows that
i^—n i i—n i i—is-^
yew^ yeN^ceC e'ec ceCyeN% c'eC ceC c'eC
$(c'-c)=y $(c'-c)=y
and hence we have
— ZyeW^^Ay — 2yeW^ |^1
ßAyXAy'
Therefore
Now we are ready to formulate the bound.
Theorem 4.2. (SDP bound) For any positive integer n and set S £ Nfi such that (n, 0,..., 0) e S
1. Bannai, E. Algebraic Combinatorics I: Association Schemes, Benjamin [Text] / E. Bannai, T. Ito. - Cummings, Menlo Park, ca, 1984.
2. Delsarte, Ph. Bounds for unrestricted codes, by linear programming [Text] // Philips Res. Reports. - Vol. 27. - 1972, June. - P. 272-289.
3. Delsarte, Ph. An algebraic approach to the association schemes of coding theory [Text] // Philips Res. Reports Supplements.- 1973. - No. 10.
4. Gijswijt, D. New upper bounds for nonbiliary codes based on the Terwilliger algebra and semidefinite programming [Text] / D. Gijswijt, A. Schrijver, H. Tanaka // Journal of Comb. Theory, Ser. A. - Vol. 113. - 2006. - P. 1719-1731/
5. MacWilliams, F.J. The Theory of Error-Correcting Codes [Text] / F.J. MacWil-liams, N.J. A. Sloane. - Amsterdam : North-Holland, 1977.
6. Mounits, B. Upper Bounds on Codes via Association Schemes and Linear Programming [Text] / B. Mounits, T. Etzion, S. Litsyn. - Advances in Mathematics, of Communications. - Vol. 1. - No. 2. -2007, May. - P. 173-195.
subject to the constraints
^A(n,o 0) — 1,
xA = o if {r(A), c(A), p(A)}£S,
REFERENCES
7. Schrijver, A. New code upper bounds from the Terwilliger algebra and semidefinite programming [Text] // IEEE Trans, on Inform. Theory. - Vol. 51. - 2005, Aug. - P. 28592866.
Б. Маунитс
НОВЫЕ ВЕРХНИЕ ГРАНИЦЫ НЕБИНАРНОГО КОДА
Статья посвящена изучению верхних границ небинарных кодов. Границы можно получить путем линейного или полуопределенного программирования.
границы, коды, программирование.
