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AAECC 8, 415—420 (1997)
On New Perfect Binary Nonlinear Codes
A. C. Lobstein1, V. A. Zinoviev2
1Centre National de la Recherche Scientifique, Ecole Nationale Superieure desTelecommunications, 46 rue Barrault, F-75634 Paris cedex 13, France(e-mail: [email protected])2Institute for Problems of Information Transmission, Russian Academy of Sciences,Bol’shoi Karetnyi, 19, GSP-4, Moscow, 101447, Russia (e-mail: [email protected])
Dedicated to Aimo ¹ieta( va( inen on the occasion of his 60th birthday
Received: November 26, 1996; revised version: March 14, 1997
Abstract. We present a new construction of binary nonlinear perfect codes withminimum distance 3 and lowerbound the number of nonequivalent such codes.
Keywords: Generalized concatenated codes, Binary nonlinear perfect codes.
1 Introduction
A block code C over field Fq
(whose elements we denote 0, 1, . . . , q!1), withlength n, size qk (k71), and minimum distance d is denoted by (n, k, d ). If C islinear, it is denoted by [n, k, d]. For any a3Fn
q, the set C#a"Mc#a : c3CN is
a coset of C. A q-ary code C with minimum distance d"2e#1, length n, and sizeqk is perfect if and only if :
qk ·e+i/0
(q!1)iAn
i B"qn.
It is known ([8], [9], [13], [14]) that a nontrivial perfect code C over Fq
necessarily has the same parameters as either a Hamming code (n"(qm!1)/(q!1), n!m, 3) or the binary [23, 12, 7] or ternary [11, 6, 5] Golay codes. Twobinary codes C
1and C
2are called equivalent if one can be obtained from the other
by permutations of positions and translations. The problem of evaluating thenumber of nonequivalent binary perfect Hamming codes is still open (see, amongnumerous papers, [10], [4], [5], [3], [7], [1]). Here, using the idea of general-ized concatenation, we describe a general construction of perfect nonlinear codes
Correspondence to: A. C. Lobstein
and then bound the number of nonequivalent codes obtained through thisconstruction.
2 Perfect Codes by Concatenation
Our construction is inspired by generalized concatenated codes, which were intro-duced by Zinoviev [11] and are also the basis of the construction in [12]. LetnA"2u and n
B"2m. Let A
1be a binary extended perfect code (n
A, n
A!1!u, 4),
A2
be an nB-ary (n
A, n
A!1, 2) code, and A
3be a q
3-ary (n
A, n
A, 1) code, where
q3"2nB~1~m. Let B be a binary (n
B, n
B, 1) code which can be represented as an
union,
B" Zi/0,1
Bi,
where, for i"0, 1, Biis an (n
B, n
B!1, 2) code. Consider for each B
ithe partition
Bi"
nB~1Zj/0
Bi, j
, (2.1)
where, for j"0, 1, . . . , nB!1, the B
i,j’s are disjoint binary (n
B, n
B!1!m, 4)
codes. The Bi,j
’s partition B, hence any b3B belongs to exactly one Bi,j
and, ifb has index k in B
i,j, we see that (i, j, k)3M0, 1N]M0, . . . , n
B!1N]M0, . . . ,
2nB~1~m!1N characterizes vector b; we note: b"b (i, j, k).Consider, for l"1, 2, 3, a codeword a(l)"(a(l)
1, . . . , a(l)
nA)3A
l. For s"1, . . . , n
A,
the triple (a(1)s
, a(2)s
, a(3)s
) designates a codeword b"b (a(1)s
, a(2)s
, a(3)s
)3B. Definea new code
C"M (b(a(1)1
, a(2)1
, a(3)1
) D . . . Db (a(1)nA
, a(2)nA
, a(3)nA
)) : a(l)3Al, 16l63N.
Codes Ai, B and C are called, respectively, outer, inner and concatenated codes.
Theorem 2.1 C is a binary extended perfect (n"2m`u, n!(m#u)!1, 4) code.
Special cases of this basic construction were obtained in [12] and [6]. In turn thisconstruction is a special case of a construction by Phelps [5], described ina different way, without mentioning concatenation construction.
3 Modified Concatenation Constructions
First we note that the extended perfect binary code which we denote by B0,0
hasexactly n
Bcosets of weight 1: they are all codes B
1, j, j"0, . . . , n
B!1. Denote by
v(h) the binary vector of length nB
whose only nonzero element is on its h-thposition: any subcode B
1,jis uniquely defined by some vector v(h). Without loss of
generality, we can assume that h"j. Further, let the code B0, j
with j'0 bedefined by the vector w(j) with two nonzero positions, 0 and j. Therefore, partition(2.1) can be expressed as follows:
B0"B
0,0ZG
nB~1Zj/1
(B0,0
#w(j))H ; B1"
nB~1Zj/0
(B0,0
#v(j)).
416 A. C. Lobstein, V. A. Zinoviev
Denote by SB
the symmetric group of order nB! (i.e., the permutations of
M0, 1, . . . , nB!1N). Suppose that we numbered all the elements of S
B. Denote by
qi3S
Bthe permutation of ‘‘even’’ subcodes and by n
j3S
Bthe permutation of ‘‘odd’’
subcodes (i, j"1, 2, . . . , nB! ). It is clear that any permutation n
jof subcodes
B1, i
can be expressed as a permutation of vectors v(i) defining the cosets. Similarly,any permutation q
iof subcodes B
0,j( j71) can be expressed as a permutation of
vectors w(j).Let G"M1, . . . , n
AN be the set of coordinate positions of A
2. Denote by E
i"
(Ei,1
, . . . , Ei,k
) the i-th partition of G for even subcodes and byO
j"(O
j,1, . . . , O
j,s) the j-th partition of G for odd subcodes. Two partitions of
G are said to be equivalent if they differ only by the order of their elements.It is known (see, e.g., reference in [13]) that any binary (extended) perfect code is
self-complementary. Assume that for any pair of complementary vectors (a, a6 )3A1,
the vector a has 0 on its first position. For any pair (a, a6 )3A1
we choose twoarbitrary partitions:
E (a, a6 )"E (a)"(E1(a), . . . , E
k(a)) and O(a, a6 )"O (a)"(O
1(a), . . . , O
s(a))
and two tuples of arbitrary integers (i1(a), . . . , i
k(a)), ( j
1(a), . . . , j
s(a)) such that
16i1(a)(· · ·(i
k(a)6n
B! and 16j
1(a)(· · ·(j
s(a)6n
B!.
Now we define a code
C(ME(a(1)); i1(a(1) ), . . . , i
k(a(1)); O (a(1)); j
1(a(1)), . . . , j
s(a(1) ) : a(1)3A
1N)
"M(b(a(1)1
, a(2)1
, a(3)1
) D · · · Db (a(1)nA
, a(2)nA
, a(3)nA
)) : a(l)3Al, 16l63N, (3.2)
where the encoding function b( . , . , . ) now is more complicated: for j"1, . . . , nA,
b(a(1)j , a(2)
j , a(3)j )"G
b (a(1)j , q
it (a(1)) (a(2)j ), a(3)
j ) if a(1)j "0 and j3E
t(a(1) ),
b (a(1)j , n
je (a(1)) (a(2)j ), a(3)
j ) if a(1)j "1 and j3O
e(a(1) ).
(3.3)
We see that if all partitions for all pairs (a, a6 )3A1
are trivial, i.e., ifE(a)"O(a)"(G) and q
i1(a) and nj1(a) are identical permutations, then it gives
exactly the construction described in the previous section. Obviously the para-meters of this new code are the same as C, since the permutations of the symbols ofthe alphabet of A
2do not change the distances between its codewords. So we have
the following statement.
Theorem 3.1 For any partitions E(a), O(a) of G"M1, 2, . . . , nAN and for any
integers i1(a), . . . , i
k(a), j
1(a), . . . , j
s(a), where 16i
1(a)( · · ·(i
k(a)6n
B!,
16j1(a)( · · ·(j
s(a)6n
B! and a runs over all codewords of A
1, the resulting code
C"C (ME (a); i1(a), . . . , i
k(a); O (a); j
1(a), . . . , j
s(a) : a3A
1N)
is a binary extended perfect code with d"4.
Denote by C(a) a subcode of C obtained by fixing a pair (a, a6 ) of A1:
C" Z(a, a6 )3A
1
C(a).
On New Perfect Binary Nonlinear Codes 417
Each C (a) is defined by two partitions E (a) and O(a) and two tuples of integersi1(a), . . . , i
k(a) and j
1(a), . . . , j
s(a). Our aim is to count the number M(C) of
mutually nonequivalent codes C. In order to do it, first we want to count thenumber M (a) of mutually nonequivalent codes C(a). Now we need a simpleobservation, which we formulate as a lemma. Denote by ¹q (respectively, ¹n) a setof permutations of even subcodes B
0,j(respectively, odd subcodes B
1,j) induced by
all possible translations in FnB2
.
Lemma 3.1 ¸et ¹rdenote a regular permutation representation of the abelian group
type (2] · · · ]2) of order nB. ¹hen ¹q"¹n"¹
r.
From now on, we assume that nB"4. Denote by ¸ the set of the following
forbidden partitions of G"M1, 2, . . . , nAN: E"(E
1, . . . , E
k)3¸ if and only if the
following two properties are satisfied: (i) k6nB"4; (ii) all the numbers
DE1D, . . . , DE
kD are even.
As a criterion for nonequivalence of codes, we use the following simpleresult.
Lemma 3.2 ¸et C1
and C2
be two binary codes with the same parameters, anddistance d(C
1)"d(C
2)73. Suppose that for any two positions i, j, iOj, there exist
codewords a1, a
23C
1, b
1, b
23C
2such that a
1"b
1, a
2"b
2#v(i)#v(j), and
a1, i
Oa1,j
. ¹hen C1
and C2
are not equivalent.
Proof. We cannot obtain C2
from C1
using any combination of translations andpermutations of positions. The only possibility is if they coincide. But this contra-dicts d(C
1)'2. K
Note that code C (a) is defined by two partitions and two permutations, foreven and odd subcodes. But when we consider the equivalence of two codes C(a)and C(b), it is clear that only one type of partitions and permutations is necessary(for example, even partitions and even permutations). Now as a direct corollary ofthe previous two lemmas, we have
Lemma 3.3 ¹wo codes C(a) and C(b) defined by the partitions E (a)"(Ei1, . . . , E
ik)
and E@ (b)"(Ej1, . . . , E
js) and the permutations q
i1, . . . , q
ikand q
j1, . . . , q
js, respect-
ively, are equivalent if and only if either
(i) E"E @ and (i1, . . . , i
k)"( j
1, . . . , j
s), or
(ii) E, E @3¸ and all permutations qi1, . . . , q
ik, q
j1, . . . , q
js3¹
r.
Denote by Na the set of all mutually nonequivalent codes C(a).
Lemma 3.4. ¹he cardinality of Na is at least
M (a)7AnA#23
23 B!3+i/0A(n
A!2)/2
i BA4
i B. (3.4)
Proof. If we have no limitations for the partitions and the permutations we use (seeLemma 3.3), then the number of all possible codes is equal to (nA`23
23). So we have to
count the number of forbidden codes (equivalent to each other). The cardinality of
418 A. C. Lobstein, V. A. Zinoviev
¸ is easy to count:
D¸ D"3+i/0A(n
A!2)/2
i B.Now we have to take into account the number of possible ways to choose thepermutations from the set ¹
rfor the partition consisting of i subsets. Here
¹rconsists of 4 permutations, because n
B"4. So for the set with i subsets we have
(4i) such choices, yielding (3.4). K
Now using M(a) of the nonequivalent codes Ci(a), we build the nonequivalent
extended perfect codes C. Such a code consists of 2nA~2~u codes Ci(a). Note that if
two codes Ci(a) and C
j(b) are nonequivalent, then for any binary vector c of length
nA, C
i(a) and C
j(b#c) are also nonequivalent. Let us call shortly the code
Cj(b#c), which is equivalent to C
j(b) (as well as to C
j(c)), a code of type C
j(b). We
define now the following class of mutually nonequivalent codes NC. This class isa union of sets NC
s, s"1, 2, . . . , where NC
sconsists of the codes of s-th type. For
simplicity, let M(s)"DNCsD and !"2nA~2~u. ¹he codes of s-th type NC
sconsist of
t1codes of type C
1(a
1), t
2codes of type C
2(a
2), and so on . . . , t
s"!!t
1! · · · !
ts~1
codes of type Cs(a
s), where t
1, . . . , t
s~1are arbitrary integers the sum of which is
less than !, C1(a
1), . . . , C
s(a
s)3Na, and all C
i’s are distinct. ¹hen we have
M (s)"AM(a)
s BA!!1
s!1 B , where s6minMM(a), !N.
For the case M(a)7!/2, we clearly have the following lower bound.
Theorem 3.2 ¸et the value M(a) in ¸emma 3.4 be such that M (a)72nA~3~u, wherenA"2u74. ¹hen our construction provides at least M(C) binary extended perfect
nonlinear codes of length n"4nA, where
M (C)'22c (n~-0'2n) and c'0.25.
For nB"n
A"4, the outer code A
1consists of two codewords at distance 4.
Therefore any code C (a) is an extended perfect code of length 16. So Lemma 3.4gives the following lower bound on the number of nonequivalent extended perfectnonlinear codes of length 16:
M(C)7A4#24!1
24!1 B!(24#6)"17520.
Note that Phelps [4] showed that there are at least 31021 nonequivalent perfectcodes of length 15. It means that our estimation is quite rough.
For nA"8, we have according to Lemma 3.4 that M(a)77 888 690, which
gives the following lower bound for the number of nonequivalent binary extendedperfect codes of length n"32: M (C)72167.98.
Similarly, for n"64, we have M(a)737 711 260 695, and M(C)7227208.49.It is clear that for arbitrary n
B'4, the value M(a) is approximately equal to
(nB!`nAnA
). Then in the best case, when the values nA
and nB
are growing so thatnB!Kn
A, our estimation gives only the following number of nonequivalent ex-
tended perfect codes of length n"nAnB: M (C)722cn where c'1/n
B. This means
On New Perfect Binary Nonlinear Codes 419
again that our estimation is too rough. Note that in the recent paper [1] thefollowing lower bound for the number of different (not nonequivalent) perfectcodes of length n was obtained:
22n`12 ~-0'(n`1)
· 62 n`5
4 ~-0' (n`1)· (1!o (1)).
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420 A. C. Lobstein, V. A. Zinoviev
