New non-binary code bounds based on a parity argument
arXiv:1606.05144v1 [math.CO] 16 Jun 2016
Sven Polak1
Abstract. For q, n, d ∈ N, let Aq (n, d) be the maximum size of a code C ⊆ [q]n with minimum distance at least d. We give a parity argument that yields the new upper bounds A5 (8, 6) ≤ 65, A4 (11, 8) ≤ 60 and A3 (16, 11) ≤ 29. These in turn imply the new upper bounds A5 (9, 6) ≤ 325, A5 (10, 6) ≤ 1625, A5 (11, 6) ≤ 8125 and A4 (12, 8) ≤ 240. Furthermore, we prove that, if q | n, then there is a 1-1-correspondence between symmetric (n/q, q)-nets (which are certain designs) and codes C ⊆ [q]n of size qn with minimum distance at least n(q − 1)/q. We derive the new upper bounds A4 (9, 6) ≤ 120 and A4 (10, 6) ≤ 480 from these ‘symmetric net’ codes. Keywords: code, nonbinary code, upper bounds, Kirkman system, parity, symmetric net. MSC 2010: 94B65, 05B30.
1
Introduction
Fix n, q ∈ N. A word is an element v ∈ [q]n := {0, 1, . . . , q − 1}n .2 For two words u, v ∈ [q]n , we define their (Hamming) distance dH (u, v) to be the number of i with ui 6= vi . A code is a subset of [q]n . For any code C ⊆ [q]n , the minimum distance dmin (C) of C is the minimum distance between any pair of distinct code words in C. Now we define, for d ∈ N, Aq (n, d) := max{|C| | C ⊆ [q]n , dmin (C) ≥ d}.
(1)
Computing Aq (n, d) and finding upper and lower bounds for it is a long-standing research interest in combinatorial coding theory (cf. MacWilliams and Sloane [13]). In this paper we find (for some q, n, d) new upper bounds on Aq (n, d), based on a parity-argument. In some cases, it will sharpen a combination of two well-known upper bounds on Aq (n, d). Theorem 1.1 (Well-known upper bounds on Aq (n, d)). Fix q, n, d ∈ N. (i) If qd > (q − 1)n, then Aq (n, d) ≤
qd . qd − n(q − 1)
(2)
This is the q-ary Plotkin bound. (ii) We have that Aq (n, d) ≤ q · Aq (n − 1, d). Proof. A proof of these statements can be found in [4]. Plotkin’s bound can be proved by comparing the leftmost and rightmost terms in (3) below. The second bound follows from the observation that in a (q, n, d)-code any symbol can occur at most Aq (n − 1, d) times at the first position. 1
Korteweg-De Vries Institute for Mathematics, University of Amsterdam. E-mail:
[email protected]. The research leading to these results has received funding from the European Research Council under the European Unions Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement №339109. 2 We will write [q] := {0, . . . , q −1}, but for symbols other than q we use the common notation [n] := {1, . . . , n}.
1
In our parity or division arguments, we will use the following observations. Let C be a (q, n, d)code of size M . We can view C as an M × n matrix with the words as rows. For j = 1, . . . , n, let cα,j denote the number of times symbol α ∈ [q] appears in column j of C. For any two words u, v ∈ [q]n , we define g(u, v) := n − dH (u, v). Then n X X X m m−1 cα,j M ≥ n · (q − r) +r , (3) g(u, v) = (n − d) ≥ 2 2 2 2 {u,v}⊆C u6=v
j=1 α∈[q]
where 0 ≤ r < q and m := dM/qe satisfy qm − r = M . The first inequality in (3) holds because n − d ≥ g(u, v) for all u, v ∈ C, the equality is obtained by counting the number of equal pairs of entries in the same columns of C in two ways and the second inequality follows from the convexity of the binomial function. If, for some q, n, d and M , the left hand side equals the right hand side in (3), we have equality throughout. In that case it holds, for any (q, n, d)-code C of size M , that (i) g(u, v) = n − d for all u, v ∈ C with u 6= v, i.e. C is equidistant, and (ii) for each column Cj of C, there are q − r symbols in [q] that occur m times in Cj and r symbols that occur m − 1 times in Cj . In the next sections we will use (i) and (ii) to give (for some q, n, d) new upper bounds on Aq (n, d), based on division or parity arguments. Furthermore, we will prove that, if q | n, then there is a 1-1-correspondence between symmetric (n/q, q)-nets (which are certain designs) and (q, n, n(q − 1)/q)-codes C with |C| = qn. We derive some new upper bounds from these ‘symmetric net’ codes. Also, we restate a connection between generalized Hadamard matrices and codes (see Mackenzie and Seberry [12]), giving known bounds that are not contained in the ¨ most recent tables of code bounds (cf. Bogdanova, Brouwer, Kapralov, Osterg˚ ard [4, 5, 6]). Aq (n, d) A5 (8, 6) A5 (9, 6) A5 (9, 7) A5 (10, 6) A5 (10, 8) A5 (11, 6) A4 (9, 6) A4 (10, 6) A4 (10, 7) A4 (11, 8) A4 (12, 8) A4 (12, 9) A3 (16, 11)
lower bound [4, 5, 6] 45 135 41 625 28 3125 64 256 40 34 128 26 18
upper bound [4, 5, 6] 75 375 50 1855 50 8840 128 496 80 64 242 48 30
lower bound [12]
new upper bound
50
65 325
50 1625 50 8125 120 480 48 48
60 240
48 29
Table 1: An overview of the results obtained and discussed in this paper. The upper bounds are new. The exact values A5 (10, 8) = 50 and A4 (12, 9) = 48 are already known (see [12]), but they are not contained in the most recent tables of code bounds [4, 5, 6].
2
Kirkman triple systems and A5 (8, 6).
In some cases, the best known upper bounds on Aq (n, d) can be improved using parity or division arguments. In this section we consider the case (q, n, d) = (5, 8, 6), for which the best known 2
upper bound3 was 75. The bound can be pushed down to A5 (8, 6) ≤ 65. First we observe that the Plotkin bound (Theorem 1.1) gives that A5 (7, 6) ≤ 15 and hence A5 (8, 6) ≤ 5 · 15 = 75. Since, for (q, n, d) = (5, 7, 6) and M = 15, the left hand side equals the right hand side in (3), any (5, 7, 6)-code C of size 15 is equidistant and each symbol appears exactly 3 times in any column of C. If, in a (5, 8, 6)-code C, one symbol is contained 15 times in column j, the corresponding rows (and columns {1, . . . , 8} \ {j}) form a (5, 7, 6)-code of size 15. We will call these 15 rows of C a 15-block. More generally: Definition 2.1 (k-block). Let C be a (q, n, d)-code in which a symbol α ∈ [q] is contained exactly k times in column j. The k × n matrix B formed by the rows of C that have symbol α in column j is called a k-block (for column j). In that case, columns [n] \ {j} of B form a (q, n − 1, d)-code of size k. Since a (5, 7, 6)-code of size 15 is equidistant, all distances in a (5, 8, 6)-code C of size 75 need to be even. To see this, note that the division into 15-blocks of C for each column is (15, 15, 15, 15, 15). Either two words are contained together in some 15-block (hence their distance is 6) or there is no column for which the two words are contained in a 15-block (hence their distance is 8). The non-existence of a (5, 8, 6)-code of size 75 follows from the following proposition. Proposition 2.1 (Parity argument). Suppose that C is a (5, 8, 6)-code containing a 15-block B for some column j ∈ [n]. Let u ∈ C \ B. Then there exists w ∈ B with dH (u, w) odd. Proof. By renumbering the symbols in each column of C, we can assume that u = 1 := 11111111. The number of times symbol 1 appears in B is exactly 7 · 3 = 21, which is odd. Therefore there must be a row in B containing an odd number of 1’s, proving the proposition. At this point we know A5 (8, 6) ≤ 74. We can pursue this idea further, by comparing lower bounds and upper bounds on the number of so-called odd pairs in a (5, 8, 6)-code. Definition 2.2 (Odd pair). Let u, v ∈ [q]n . If dH (u, v) is odd, we call {u, v} an odd pair. For any code C ⊆ [q]n , we write X := {{u, v} ⊆ C | u 6= v and dH (u, v) is odd}.
(4)
Note that a (5, 8, 6)-code C of size M that contains a 15-block has |X| ≥ M − 15, by Proposition 2.1. But we can derive more. In order to give bounds on |X|, we will use information about 15-blocks and 14-blocks. First we show that every 14-block can be obtained by removing one row from a 15-block. Proposition 2.2 (14-blocks and 15-blocks). Any (5, 7, 6)-code C of size 14 can be extended to a (5, 7, 6)-code of size 15. Proof. Since, for M = 14, the leftmost term in (3) equals the rightmost term, C is equidistant and for each j ∈ [7] there exists a unique βj ∈ [q] with cβj ,j = 2 and cα,j = 3 for all α ∈ [q]\{βj }. We can define a 15-th codeword u by putting uj := βj for all j = 1, . . . , 7. We claim that C ∪{u} is a (5, 7, 6)-code of size 15. To establish the claim we must prove that dH (u, w) ≥ 6 for all w ∈ C. Suppose that there is a word w ∈ C with dH (u, w) < 6. We can renumber the symbols in each column of C such that w = 1. Then there are two column indices j1 and j2 with uj1 = 1 and uj2 = 1. Then C \ {w} contains at most 1 + 1 + 5 · 2 = 12 times symbol 1 (since in columns j1 and j2 there is precisely one 1 in C \ {w}). But in that case, since |C \ {w}| = 13 > 12, there is a row in C that contains zero times symbol 1, contradicting the fact that C is equidistant. 3
The Delsarte bound [8] on A5 (8, 6), the bound based on Theorem 1.1 and the semidefinite programming bound (which originates from [16]) based on quadruples of code words (see [11]) all equal 75.
3
We will use the above observations to obtain a lower bound on the number of odd pairs in a (5, 8, 6)-code. Define a b f (a, b) := (3a + b)(65 − 15a − 14b) + 3 · 15 + 14 + 3 · 14ab (5) 2 2 − 2 · 21a − 8b + 1{b>0 and a=0} (65 − 14 − 39). Proposition 2.3 (Lower bound on |X|). Let C be a (5, 8, 6)-code of size 65 and let j ∈ [n]. Let a and b be the number of symbols that appear 15 and 14 times (respectively) in column j. Then the number |X| of odd pairs in C is at least f (a, b). Proof. Consider a (5, 7, 6)-code C 0 of size 15 or size 14 and define D := {u ∈ [5]7 | dH (w, u) ≥ 5 ∀ w ∈ C 0 }.
(6)
α(u) := |{w ∈ C 0 : dH (u, w) = 6}|.
(7)
For any u ∈ D, define
Then if |C 0 | = 15, then
if |C 0 | = 14, then
|{u ∈ D | α(u) = 0}| = 0,
|{u ∈ D | α(u) = 0}| ≤ 8,
|{u ∈ D | α(u) = 1}| ≤ 21,
|{u ∈ D | α(u) ≤ 1}| ≤ 39.
(8)
|{u ∈ D | α(u) = 2}| = 0. This can be checked fast with a computer4 , by checking all possible (5, 7, 6)-codes of size 15 and 14 up to equivalence. Here we note that a (5, 7, 6)-code C 0 of size 15 corresponds to a solution to Kirkman’s school girl problem originally posed by Kirkman in 1850 in [10]: “Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily so that no two shall walk twice abreast.” Indeed, by viewing the fifteen row indices of any (q, n, d) = (5, 7, 6)-code of size 15 as ‘girls’ and the seven column indices as ‘days’, these codes are in 1-1–correspondence with solutions to the above problem (Kirkman systems) by the rule: girls i1 and i2 walk in the same triple on day j ⇐⇒ Ci01 ,j = Ci02 ,j ,
(9)
0 stands for the i, j-th entry of C 0 . The minimum distance 6 condition then corresponds where Ci,j to the condition that no two girls shall walk twice abreast, since g(u, v) ≤ 1 for all u, v ∈ C 0 with u 6= v. So to establish (8), one needs to check all Kirkman systems (there are 7 nonisomorphic Kirkman systems [7]) and all possible (5, 7, 6)-codes of size 14, of which there are at most 7 · 15 by Proposition 2.2. Let G = (C, X) be the graph with vertex set V (G) := C and edge set E(G) := X. Consider a 15-block B determined by column j. By (8), each u ∈ C \ B has ≥ 1 neighbour in B and all but ≤ 21 elements u ∈ C \ B have ≥ 3 neighbours in B. So by adding ≤ 2 · 21 new edges, we obtain that each u ∈ C \ B has ≥ 3 neighbours in B. Similarly, for any 14-block B determined by column j, by adding ≤ 8 new edges we achieve that each u ∈ C \ B has ≥ 1 neighbour in B. Hence, by adding ≤ (2 · 21 · a + 8 · b) edges to G, we obtain that a b |E(G)| ≥ (3a + b)(65 − 15a − 14b) + 3 · 15 + 14 + 3 · 14ab. (10) 2 2
This results in the required bound, except for the term with the indicator function. That term can be added because |{u ∈ D | α(u) ≤ 1}| ≤ 39 if |C 0 | = 14, by (8). 4
All computer tests in this paper are small and can be executed within a minute on modern personal computers. Without a computer it is not hard to prove the less strict bound A5 (8, 6) ≤ 70. See Theorem 3.1.
4
It is also possible to give an upper bound on |X|. If C is a (5, 7, 6)-code of size t, the number of pairs {u, v} ⊆ C with dH (u, v) = 7 is at most the leftmost term in (3) minus the rightmost term in (3). The resulting values h(t) are given in Table 2. t h(t)
15 0
14 0
13 1
12 3
11 6
10 10
9 8
8 7
7 7
6 8
5 10
Table 2: Upper bound h(t) on the number of pairs {u, v} ⊆ C with dH (u, v) = 7 for a (5, 7, 6)code C with |C| = t. Theorem 2.4 (A5 (8, 6) ≤ 65). Suppose that C is a (q, n, d) = (5, 8, 6)-code with |C| = 65. Then each symbol appears exactly 13 times in each column of C. Hence, A5 (8, 6) ≤ 65. (j)
Proof. Let ak be the number of symbols that appear exactly k times in column j of C. Then P (j) the number of odd pairs that have the same entry in column j is at most 15 k=5 ak · h(k). It follows that |X| ≤ U :=
8 X 15 X
(j)
ak · h(k).
(11)
j=1 k=5
One may check that if a, b ∈ Z15 ≥0 , with f (a15 , a14 ) ≤ f (b15 , b14 ) 6= 0, then
P
k
ak k = 65,
P
k bk k
= 65,
P
k
ak = 5,
P
k bk
= 5, and
15 X (7ak + bk ) · h(k) < f (b15 , b14 ).
(12)
k=5 (j)
(j)
(1)
(1)
By permuting the columns of C we may assume that maxj f (a15 , a14 ) = f (a15 , a14 ). Hence (1) (1) if f (a15 , a14 ) > 0, then ! 8 X 15 8 15 X 1 X X (j) (j) (1) ak · h(k) = U= 7ak + ak · h(k) (13) 7 j=1 k=5
