Upper bounds on |C2| for a uniquely decodable code pair
(C1,C2) for a two-access binary adder channel
Citation for published version (APA):
Tilborg, van, H. C. A. (1983). Upper bounds on |C2| for a uniquely decodable code pair (C1,C2) for a two-access binary adder channel. IEEE Transactions on Information Theory, 29(3), 386-389.
https://doi.org/10.1109/TIT.1983.1056667
DOI:
10.1109/TIT.1983.1056667 Document status and date: Published: 01/01/1983
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Upper Bounds on Ic, 1 for a Uniquely
Decodable Code Pair (C,, C,i
for a Two-Access Binary
Adder Channel
HENK G. A. VAN TILBORG, MEMBER, IEEE
DEDICATED TO JESSIE MACWILLIAMS ON THE OCCASION OF HER RETIREMENT FROM BELL LABORATORIES
Abstract-
An algebraic and a combinatorial upper bound are derived on IC,( given the code C,, where (C,, C,) is a uniquely decodable code pair (C, , C, ) for a two-access binary adder channel. A uniquely decodable code with rate pair (0.5 170, 0.78 14) is also described.I. INTRODUCTION
C
ONSIDER a binary two-access adder channel. The two users use a binary block code C, , respectively C,, and we shall assume that they are in bit and block synchro- nization. In the’noiseless case (which we shall discuss here) the messages c, E C,, and c2 E C, will be received as cl + c2, where the addition + takes place in Z.This noiseless two-access channel has been studied by several authors, e.g., Liao [5], Ahlswede [l], Kasami, Lin et al. [2], [3], [4], [6], [9], and van Tilborg [8].
Liao [5] has shown that the capacity region of this channel can be described by
O<R, < 1, O<R, < 1,
R, -I- R, < 3/2. (1) The code pair (C,, C,) is called uniquely decodable if the sums c, + c2 of all pairs (c,, c2) E C, X C, are all differ-
ent. This means that the receiver can uniquely determine the codewords c, and c2 from their sum.
In van Tilborg [8] it is shown by combinatorial methods that if n is the length of the uniquely decodable code pair cc,, G>
(4
From (2) one can show with elementary asymptotic meth-Manuscript received February 8, 1982; revised July 15, 1982. This work was presented at the 1982 IEEE International Symposium on Information Theory, Les Arcs, France, June 21-25.
The author is with the Department of Mathematics and Computing Science, Eindhoven University of Technology, Eindhoven, The Nether- lands.
ods that the rate pair (R,, R2) of any uniquely decodable code pair (C,, C,) satisfies (1).
In Wei, Kasami, Lin, and Yamamura [9] the existence of certain good uniquely decodable codes is demonstrated. Asymptotic methods lead to the lower bound in Fig. 1.
In the same paper the authors associate a certain graph I(C,) with the code C,. They show that determining the maximal size of a code C, such that (C,, C,) is uniquely decodeable is equivalent to determining the maximal size coclique in I( C,).
Determining maximal cocliques (or their size) soon be- comes computationally infeasible for larger values of n. We propose an easy to compute manner of upper bounding ]C,] given C,. In this way one again can derive (2). More- over by taking a closer look at these bounds one often can improve them in particular cases. We also find a uniquely decodable code (C,, C,) with parameters n = 5, IC,I = 6, ]C,( = 15. This leads to a rate pair above the time sharing line in Fig. 1 formed by the two codes {(O,O), (1, l)} and ((0, o>, a 11, (40)).
II. DEFINITIONS
Let V, denote (0, l>” and let $ denote modulo 2 addi- tion. An easy way of describing codes (subsets of V,) and certain properties of codes uses the terminology of group algebras.
Definition: The group algebra (C( V,), q , *) of V, over Q: is the set of formal sums E, E v uUz”, with addition EE and multiplication * defined by ’
(i&O’) q ( u!vb.r’) = $‘au + hub”, (3)
(~~u”z~)*(~~~‘~~z~)=~~~(~~=~~“.b.jz~.
n n ”
(4)
A subset A of V, can now be denoted by the element E UcA~” of C(V,). For this element we shall use the same 001%9448,‘83/0500-0386$01.00 01983 IEEE
VAN TILBORG: UPPER B O U N D S O N 1 C, 1 387
0.5
t,, I, I, I, II,
0 0.5 I .o 5
Fig. I. Lower bound on achievable rates of uniquely decodable codes.
(See [91.)
letter A. O f particular interest will b e the sets
Yk: = {u E v,Iw,&) = k}, O<k<n, (5) where w, denotes the H a m m ing weight.
For the following notions a n d lemmas the reader is referred to M a c W illiams a n d Sloane [7].
L e m m a I: T h e characteristic numbers B,, 0 < k, Q n, of a code C in V, defined by B, = ICI-2 c UE Y, satisfy 2 Xc’- 1) u,c, + ‘.. + U”C,, (6) ‘,~04Pk(n, I), (7) B, = ICJ
where A, is the 1 th coefficient of the distance enumerator of C a n d where the Krawtchouk polynomial Pk( n, x) is given by
Pk(ny
x) = 5
(-2)‘( i 1ij( 7).
i=O
(8)
Definition 2: T h e annihilator polynomial a(x) of the code C with characteristic numbers B, is given by
a(x) = 2 ”IcI-’
,<,<vB e,(’ - 8. . . > k (9) T h e degree r of a(x) is called the external distance of C. T h e expansion of a(x) in terms of the Krawtchouk poly- n o m ials
dx> = c akPk(n,
x>
k=O
(10)
is called the Kruwtchouk expansion of a(x). T h e coeffi- cients (Ye, 0 G k < r, are called the Krawtchouk coefficients.
Definition 3: For all x E V, a n d 0 < k < n B(x, k): = #{ c E Cld,(x, c) = k}. L e m m a 4:
(11) C* Y, = E B(x, k)z”.
XE 4, (14
T h e o r e m 5: Let (Y,, 0 < i < r, b e the Krawtchouk coef- ficients of the annihilator polynomial a(x) of a code C. T h e n
c* 6 (YkYk = v,, (13)
k=O
kcoa,B(x, k) = 1, for allx E V,. (14)
111. RESULTS
In the sequel all the numbers (Ye, B(x, k), etc., will b e defined with respect to the code C = C,.
L e m m a 6: Let (C,, C,) b e uniquely decodable. T h e n c B(c,, k) =s (3”. (15) C2~G
Remark: Note that (15) is equivalent with
I{(c,,c,) E C, x W ,( c,, c2) = k}l < (;)Zk. (16)
Proof:
It is sufficient to show that for any u E Ykl{(c, 2 c2) E c, x qc2 = c, cl+ u}I < 2k. (17) W e m a y assume without loss of generality that u has its ones in the first k coordinate places. Now assume the contrary of (17) i.e., there are more than 2k pairs (c,, c2) E C, x C, with c2 = c, @ U. By the pigeon hole principle at least two c, must agree o n the first k coordinates, say c; a n d c;‘. If o n e now considers the two pairs of codewords (c;, c;‘) = (c;, c;’ @ u) a n d (c;l, c;) = (c;‘, c; + u) in C,
x
C,,
then o n e easily verifies thatc; + c;’ = c; + (c;’ a 3 u)(S;I + (c; $ u) = c;’ + c;.
(18)
Indeed (*) holds trivially at the last n - k coordinates since ui = 0 for i > k, a n d (*) holds o n the first k coordi- nates since (c;)~ = (~;l)~ for i 6 k. Obviously (18) con- tradicts the assumption that (C,, C,) is uniquely decodable.
0 L e m m a 7: Let (C,, C,) b e uniquely decodable. T h e n
c B(c,,
k) <
(;]2”-“.(19)
C,EC,
Remark: Note again that (19) is equivalent to
Proof: The proof is very similar to the proof of Lemma 6. Again it is sufficient to prove that for any u E Y,
#{(c,,c2) E c, x c*1c2 = c, @ u} < 2”-k. (21) Again assume that u has its ones in the first k coordinate places. Assuming the contrary of (21) we now have the pairs (c;, c;) = (c;, c; $ u) and (c;I, c;‘) = (c;I, c;’ @ U) in C, x C,, where c; and c; now agree on the last n - k
coordinates. Again c; + c; = c; + (c; @ u) = c;’ + (c;l @ u) = c;’ + c;’ because on the last n - k coordinates c; and c;’ agree and on the first k coordinates c; + (cl @ U) equals the all-one vector for i = 1,2. 0
Lemmas 6 and 7 together yield another proof of (2). Indeed, summing (16) or (20) (depending on the minimum of 2k and 2n-k) for 0 < k G n one obtains
From (14) (15), and (19) one easily deduces the following theorem.
Theorem 8: Let (Y,, 0 < i < r, be the Krawtchouk coef- ficients of the annihilator polynomial a(x) of a code C,. Let (C,, C,) be uniquely decodable. Then
IC,l < i max (0, ak)( l)2”“(k,“-k).
k=O
(22) The importance of Theorem 8 lies mainly in its power to exclude large classes of candidates for the code C,, when one is looking for good uniquely decodable code pairs cc,, c2>.
Example: C, is the Preparata code of length n = 22” - 1 m > 2 and size 2n-2m-t’. It is well known (cf. [7]) that r’= 3 for’these codes and that lyo = (Y, = 1, a2 = (Ye = 3/n. It follows from Theorem 8 that
IC,l 6 4n2 - 4n + 3.
For m = 2 one finds at the best the pair (]C, 1, ]C,]) = (2’, 843) of length 15 with corresponding rate pair (R,, R2)
= (0.5334,0.6480). For m > 2 the results are even poorer. O f course if one does not know the c+‘s from the literature, then one can compute the (Y~ quite easily from the distance distribution by means of (7), (9), and (10).
If Theorem 8 leaves us with a promising candidate C,, then a closer look at Lemmas 6 and 7 will often help in eliminating C, as good candidate or in finding the code C,. The idea is that instead of using (13), we try to “cover” V, ormostof ~~byC*Y,,C*Y,;..,C*Y,,sminimal.The
following example may clarify this idea.
Example: Consider n = 5, C, = (0,3, 12,21,26,31} in binary notation. It is rather easy to check that
A(z) = (3 + 4z2 + 8z3 + 2z4 + z5)/3, B(z) = (9 + 10z2 + 16z3 + 13z4)/9, a(x) = 256-‘(1 - x/2)(1 - x/3)(1 - x/4)
= iPo(5, x) + iP,(5, x) + iP2(5, x) + jP3(5, x).
It follows from Theorem 8 that
IC,l < f + f5 x 2 + ;10 x 22 + +10 x 2* = 17. With (16) it is not difficult to obtain a smaller upper bound for ]C,]. Indeed (16) implies that ]C, fl C,] < 1 and that
IC, f-l {x E Iqd,(x,C,) = l}] d 5 x 2 = 10. It is easy to check that {x E qldH( X, C,) > l} = {6,9,22,25} (again in binary notation). Thus ]C,] < 1 + 10
+ 4 = 15. Moreover if one wishes to find a code C, with ]C,l = 15 such that (C,, C,) is uniquely decodable, then C, must include 6, 9, 22, 25 and the 1 + 5 X 2 in equalities,
which added up yield (16) for k = 0 and k = 1, must all be equalities. These form the first set of eleven equations below. The first equation simply states that ]C, n C,I < 1. The second equation states that exactly one of the three words in C, @ (1, 0, 0, 0,O) starting with a one must be in C,, etc. It also follows that each of these eleven equations must contribute one element to C,. In other words: an element of V, that occurs in more than one equation cannot be an element of C,. It follows from (2)-(9) that (1, 27, 29, 8, 2, 30, 23, 4) n C, = 0 and {13,14,17, 11,20,24,7, IS} c C,.
From (20) one can also deduce eleven equations by plugging in k = 5 and k = 4. Indeed (16) implies that
IC, n {x E T/513,,EC,[14A~~~I) = 5ljl d 1
ic,
n {x E ~13,,EC,[dHb~
4 = 4ljl G
5 x 2. As before one can easily check that the set {x E&P&C, [dH(x, c,) G 31) is the set {6,9,22,25}. So again
]C,( G 1 + 5 x 2 + 4 = 15. Assuming as before that IC,]
= 15 one finds the second set of eleven equations in Table I. The empty spaces should be read as zeros.
Equations (1) (lo), (1 l), (12), (21), (21) yield ten possi- bilities for the remaining three elements of C,: (O/31, 15, 16}, {15,21/26,19/28}, or {16,3/12,5/10}. It is a matter of simple verification that each of these ten choices yields a uniquely decodable code pair (C,, C,) of length 5, with sizes 6 respectively 15. The corresponding rate pair (R,, R2) = (0.5170,0.7814) lies above the time sharing line in Fig. 1.
IV. CONCLUSION
Theorem 8 yields an easy to compute upper bound on the size of C, for a uniquely decodable code pair (C,, C,) in terms of the Krawtchouk coefficients of the annihilator polynomial of C,.
With more combinatorial ad hoc methods, Lemmas 6 and 7 will often give improvements on these bound and/or yield information on which vectors are in C,.
The example in which these methods are demonstrated yields a rate pair (0,5 170,0.7814) which lies above the presently known lower bound on the achievable rates of uniquely decodable codes. The main problem still is how to find good candidates for C,.
VAN TILBORG: UPPER B O U N D S O N 1 C, 1 TABLE I EXAMPLE 389 ),3l, 3, 12,21,26, 5, IO, l9,28, 15, 16, I I I 11 I I I I I I I :- t i I, 13,24,27,29, 17, 11, 8, 11 I 1 I I I I I I I 1 I 1 I 11 1 1 I I I 61 I T i
t
I
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