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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 12, DECEMBER 2003
Common Coordinates in Consecutive Addresses Ning Cai, Ludo M. G. M. Tolhuizen, Senior Member, IEEE, and Henk D. L. Hollmann
An address list of length M is a sequence a(0); a(1); . . . ; a(M 0 1) of M distinct elements from Qn ; of course, we require that M q n . If S Qn , the common coordinates of S , denoted by cc(S ), is the set of positions in which the vectors of S agree, that is, cc(S )
Abstract—We consider lists of distinct -ary addresses of length . We wish that any consecutive addresses in such a list agree in many positions. We give upper bounds on what can be achieved. Moreover, for each and , we give explicit constructions of address lists, among which is the conventional -ary reflected Gray code, that attain these bounds for all simultaneously. This work has applications in address retrieval on optical disc.
=
f 2 [1
; n]
i
i = ti ]g:
s
Here and in the sequel, [i; j ] denotes the set of natural numbers between i and j (inclusive), that is,
f 2 j g For an address list A = (0) (1) . . . ( 0 1) of length and an integer , we denote by ( A) the minimum number of common coordinates of any consecutive vectors in A, that is, ( A) = minfj (f ( ) ( + 1) . . . ( + 0 1)g)j 2 [0 0 ]g 1 For each positive integer , we define ( ) = maxf ( A) j A is an address list of length g [i; j ] = a
i
x
;a
x
j
;
;a M
;
;
:
M
g b;
b
Index Terms—Address ordering, Gray codes.
j 8s2S 8t2S [
b
g b;
I. INTRODUCTION
cc
a i ;a i
a i
In [1], van Dijk et al. considered coding for an error-prone channel, where a side channel possibly informs the decoder about a part of the information that is encoded in the transmitted codeword. The encoder does not know about which part of the information, if any, the decoder will be informed. Several constructions were given of codes for which the error-correcting power is enhanced if some information symbols are known to the decoder. The above so-called coding for informed decoders can be applied in address retrieval on optical disc [2]. In this application, the decoder knows, from the system specifications and the address of the sector in which the read/write head is intended to land, a relatively small disc area in which the head will actually land. The common part of the addresses of the sectors residing in this area can be used as known information in informed decoding. Clearly, we would prefer this common part to be as large as possible. Motivated by this application, we shall study the following problem (defined more formally in Section II). Assuming that each sector address is a q -ary vector of length n, we are interested in lists of addresses such that the addresses of any b consecutive sectors agree in many positions. In Section III, we derive upper bounds on what can be obtained. A Gray code is an address list for which any two consecutive addresses differ in exactly one position, and thus is the best that can be done in the special case b = 2. In Section IV, we study Gray codes and show that they meet the upper bounds from Section III for all b simultaneously if they satisfy some extra condition. In Section V, we describe three Gray codes, among which is the well-known q -ary reflected Gray code, that meet this condition and therefore are perfectly suited to be applied in conjunction with informed decoding. In Section VI, we discuss problems related to the present work. II. PROBLEM DESCRIPTION Throughout this correspondence, the following notation and terminology will be used. We fix a positive integer n and a set Q of size q .
Manuscript received May 26, 2002; revised August 18, 2003. The material in this correspondence was presented in part at the 23rd Symposium on Information Theory in the Benelux, Louvain-la-Neuve, Belgium, May 2002. N. Cai was with the Department of Information Engineering, The Chinese University of Hong Kong, N.T. Hong Kong. He is now with the Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany (e-mail:
[email protected]). L. M. G. M. Tolhuizen and H. D. L. Hollmann are with Philips Research Laboratories, 5656 AA Eindhoven, The Netherlands (e-mail: ludo.tolhuizen@ philips.com;
[email protected]). Communicated by G. Battail, Associate Editor At Large. Digital Object Identifier 10.1109/TIT.2003.820034
b
g b; M
b
i
;M
b
:
M
g b;
M
:
We are interested in finding g (b; M ) and in address lists that attain this maximum. Note that Gray codes solve this problem for the special case b = 2. Instead of saying that vectors agree in the positions indexed by I , we will simply say that vectors agree on I . III. UPPER BOUNDS ON g (b; m)
We start with an elementary property. Any S Qn is contained in the set of the q n0jcc(S )j vectors that agree on cc(S ) with a given element from S , and so for each S Qn ; jS j qn0jcc(S)j: (1) In particular, taking for S the initial b vectors of an address list, we find that b q n0jcc(S )j , so jcc(S )j n 0 dlogq be, whence g (b; M )
for all b and M;
max(0 0 dlogq e) ;n
b
:
(2)
In the following lemma, a slightly sharper upper bound is provided. Lemma 1: Suppose l and b are such that b q l01 + 2 and M > q l .
Then g (b; M ) n 0 l 0 1. Proof: Let A = a(0); a(1); . . . ; a(M 0 1) be an address list of length M . As b q l01 + 2, it follows from (1) that the initial b vectors of A agree in at most n 0 l positions. We assume that they agree in exactly n 0 l positions (otherwise we are done), and denote their common coordinates by I . As q l < M vectors agree with a(0) on I , at least one vector in A does not agree with a(0) on I . Let j be such that the vectors a(0); a(1); . . . ; a(j 0 1) all agree with a(0) on I , but a(j ) does not. By definition I
As b 0 1
(f cc
>
0
a(j
b
0
+ 1); a(j
b
+ 2); . . . ; a(j
l01 , it follows from (1) that q a(j
0
b
+ 1); a(j
0
b
+ 2); . . . ; a(j
0 1)g)
:
(3)
0 1)
have at most n 0 l common coordinates, so the inclusion in (3) is, in fact, an equality. As a(j 0 b + 1) and a(j ) do not agree in all positions indexed by I ,
f
cc( a(j
0
b
+ 1); a(j
0
b
g
+ 2); . . . ; a(j ) )
is a proper subset of I , and so its cardinality is less than n 0 l. 1For notational convenience, the dependence of and has not been expressed.
0018-9448/03$17.00 © 2003 IEEE
on the fixed parameters
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 12, DECEMBER 2003
a(k) and a(i) agree in at least n 0 l 0 1 of their n 0 l leftmost positions
IV. GRAY CODES ACHIEVING g (b; M ) A Gray code is an address list for which any two consecutive addresses differ in exactly one position. A Gray code is an obvious choice for an address list, as then at least for small b we stand a chance of having many common coordinates in any b consecutive addresses. It is not true, however, that a Gray code necessarily meets the upper bounds from Section III, as the following example shows. Example 1: The address list 0000; 0001; 0011; 0111; 1111; 1011; 1001; 1000; 1010; 1110; 1100; 1101; 0101; 0100; 0110; 0010
is a Gray code. Its b = 5 initial vectors have no common coordinate, while the bound in Lemma 1, with l = q = 2, equals 1. In order to describe a property guaranteeing that a Gray code meets the upper bounds in Lemma 1, we need the following definition. Definition: The Gray code A = a(0); a(1); . . . ; a(M 0 1) has property P (l) if for each u and v in [0; M 0 1], a(u) and a(v ) agree in their n 0 l leftmost positions whenever b qu c = b qv c.
Lemma 2: Let A = a(0); a(1); . . . ; a(M 0 1) be a Gray code that has property P (l). Let i 2 [0; M 0 1], and let j := b qi c1 ql + (q l 0 1). Then we have that i) for each k 2 [i; min(j; M n 0 l leftmost positions;
0 1)], a(i) and a(k) agree in their
M 0 2, then for all k 2 [j + 1; min(j + ql ; M 0 1)], a(i) and a(k) agree in their n 0 l leftmost positions apart from the position in which a(j ) and a(j + 1) differ.
ii) if j
Proof:
i) Clearly, for each k 2 [i; min(j; M 0 1)], b qk c = b qi c and so a(i) and a(k) agree in their n 0 l leftmost positions, that is, part i) holds.
ii) Let us assume that j + 1 M 0 1. As A is a Gray code, a(j ) and a(j + 1) differ in exactly one position, and so they agree in at least n 0 l 0 1 of their n 0 l leftmost positions. We write for b qi c + 1, so j = ql 0 1. Let k 2 [j + 1; min(j + ql ; M 0 1)]. We have that k = q l + rk with rk 2 [0; q l 0 1], so b qk c = . Consequently, the vector a(k), that agrees with a(j +1) in its n0l leftmost positions because of property P (l), agrees with a(j ) in its n 0 l leftmost positions, except for the position in which a(j ) and a(j + 1) differ. As a(i) and a(j ) agree in their n 0 l leftmost positions, we have proved part ii). Lemma 3: Let A = a(0); a(1); . . . ; a(M 0 1) be a Gray code that has property P (l) for each l 2 [0; n 0 1]. For each b 2 [2; M ], we have that
g(b; A) =
n 0 l(b) 0 1; n 0 l(b);
if M if M
3309
ql b ql b
( )
( )
where l(b) = dlogq (b 0 1)e. Proof: For b = 2, the lemma follows by combining (2) and the definition of a Gray code. Now let b 2 [3; M ]. We denote dlogq (b 0 1)e by l. It is clear that b 2 [2 + ql01 ; 1 + ql ]. If M q l , then b qi c = 0 for all i 2 [0; M 0 1]. As A has property P (l), all addresses agree in their n 0 l leftmost positions, and so g(b; A) n 0 l. From (2) it follows that g(b; A) n 0 l. Now we assume that M > q l . According to Lemma 1, g (b; A) n 0 l 0 1. Let i 2 [0; M 0 b]. In the notation of Lemma 2, j i and
for each
k 2 [i; min(j + ql ; M 0 1)] [i; min(i + ql ; M 0 1)]: Note that Lemma 3 implies that a Gray code that has property P (l) for each l 2 [0; n 0 1] achieves the upper bound on g (b; M ) from Lemma 1. In Section V, we will present three explicit constructions of such Gray codes for all values of q and n. As a consequence, we have the following theorem. Theorem 1: For each q; M and b
g(b; M ) =
2 [2; M ], we have
n 0 l(b) 0 1; n 0 l(b);
if M > q l(b) if M q l(b)
where l(b) = dlogq (b 0 1)e. A Gray code that has property P (l) for each l 2 [0; n 0 1], as constructed in Section V, achieves g (b; M ) for all b simultaneously.
Remark 1: Suppose A = a(0); a(1); . . . ; a(q n 0 1) has property P (l). By definition, for each m 2 [0; qn0l 0 1], the ql vectors from fa(m 1 ql + i)l j i 2 [0; ql 0n1]g agree in their n 0 l leftmost positions. As there are q vectors in Q that agree in their n 0 l leftmost positions, we conclude that a(i) and a(j ) agree in their n 0 l leftmost positions if and only if b qi c = b qj c. Suppose that A has properties P (0); P (1); . . . ; P (n 0 1). Let i 2 n c. Now A has [0; q 0 1]. Let m be the largest l such that b qi c 6= b i+1 q property P (m + 1), hence a(i) and a(i + 1) agree in their n 0 m 0 1 leftmost positions. As we have seen earlier, a(i) and a(i + 1) do not agree in their n 0 m leftmost positions. Hence, a(i) and a(i + 1) differ in the (n 0 m)th leftmost position. As A is a Gray code, this is the only position where these two consecutive addresses differ. As a consequence, in the binary case (q = 2), a Gray code of length n 2 that has properties P (0); P (1); . . . ; P (n 0 1) is essentially unique. Indeed, for each i, we know the position in which we should change a(i) to obtain a(i + 1), and with a binary alphabet, only one change per position is possible. Such Gray codes hence only differ in the choice of a(0). Remark 2: A Gray code A of length M can satisfy the equality g(b; A) = g(b; M ) for all b without having properties P (0); P (1); . . . ; P (n 0 1) (or being obtained from such a Gray code by permutation of
the positions). As an example, we take q = 2 and n = 3. Remark 1 implies that there is exactly one binary Gray code with 23 = 8 addresses that has 000 as initial word and has properties P (0); P (1) and P (2). In fact, this Gray code equals
G = 000; 001; 010; 011; 111; 110; 101; 100: Let A be the address list specified by
A = 000; 001; 011; 010; 110; 100; 101; 111: Note that A is essentially different from dresses differ in three positions. It can be checked that
g(b; A) = g(b; 8) =
2;
1; 0;
G , as its first and final adif b = 2 if b = 3 if b 4.
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 12, DECEMBER 2003
V. CONSTRUCTION OF GRAY CODES WITH ADDITIONAL PROPERTIES In Section IV, we showed that a Gray code having property P (l) for each l 2 [0; n 0 1] achieves equality in the upper bounds from Section III for all values of b. In this section, we describe three such Gray codes (that coincide if q = 2, see Remark 1). For the alphabet Q we take f0; 1; . . . ; q 0 1g. The Gray codes have length q n and contain all vectors from Qn . To obtain an address list of length M < q n , one can simply take the M initial vectors from these Gray codes. A. Reflected Gray Codes A well-known way to recursively construct q -ary Gray codes is by ”reflection” as described next. Construction of reflected Gray codes We define G1 = 0; 1; . . . ; q 0 1. Let n 1 and let Gn = g (0); g (1); . . . ; g (q n 0 1) be a Gray code of length q n . The address list Gn+1 is defined as 0g (0); 0g (1); . . . ; 0g (m
0 1) 1 ( 0 1) 1 ( 0 2) . . . 0 1) 3 ( 0 1) 3 ( 0 2) . . . ;
g m
;
g m
;
1g (0); 2g (0); . . . ; 2g (m
;
g m
;
g m
(4)
Theorem 2: For each n, Gn is a Gray code of length q n that has property P (l) for each l 2 [0; n 0 1]. Proof: By induction on n, one easily shows that Gn is a Gray code. We continue by showing, also by induction on n, that Gn has property P (l) for each l 2 [0; n 0 1]. This is obvious for n = 1. In order to streamline the proof, we describe the Gray code Gn differently. For all integers n and i 2 [0; q n 0 1], we denote the ith element of Gn by gn (i). The construction of Gn+1 in (4) can equivalently be described as follows: for each i 2 [0; q 0 1] and j 2 [0; q n 0 1]
n
g +1 (i
1 n+
j)
q
n (j );
if i is even if i is odd.
n (qn 0 j 0 1);
=
ig
(5)
It is trivial that Gn has property P (0). Let l 2 [1; n 0 1], let i; j 2 n+1 0 1], and suppose that b i c = b j c = a. We write i = aql + b q q with 0 b q l 0 1, and a = uq n0l + v with 0 v q n0l 0 1. Then we have that i = uq n + vq l + b. As [0; q
vq
l + b (qn0l 0 1)ql + ql 0 1 = qn 0 1
we have that
if u is even n (vql + b); n l ugn (q 0 vq 0 b 0 1); if u is odd. Similarly, writing j = aq l + c with 0 c q l 0 1, we have
n
g +1 (i)
=
=
ug
tions as well, irrespective of the parity of u. To see the latter, note that vq
l +b l q
=
v
=
vq
l+c l q
2 [0 n 0 1], the ;q
=
q
n0l 0 (v + 1) =
n l q 0 vq 0 c 0 1 l q
q
-ary representation x(i) of
=; (xn01 (i); xn02 (i); . . . ; x0 (i));
where xj (i) 2 Q for all j; and i =
i
n01
is the
j (i)qj :
x
j =0 n It is easy to see that for all u and v in [0; q 0 1] and all l 2 [0; n 0 1] u q
l
=
l
if and only if xi (u) = xi (v ) for all i 2 [l; n 0 1]:
2 [0 n 0 1], we define n0 ( ) n0 ( ) 0 n0 ( ) . . .
For each i a(i)
v q
= (x
;q
1 i ;x
2 i
x
1 i ;
;
x2 (i)
0
x1 (i); x0 (i)
0
x1 (i))
Theorem 3: A = a(0); a(1); . . . ; a(q n 0 1) is a Gray code that has property P (l) for all l 2 [0; n 0 1]. Proof: It is clear that a(i) and a(j ) agree in their n 0 l leftmost positions if x(i) and x(j ) agree in their n 0 l leftmost positions, so property P (l) holds for all l 2 [0; n 0 1]. All we need to show is the Gray property. Now, let 0 i q n 0 2. Let r 2 [0; n 0 1] be such that x0 (i) = x1 (i) = 1 1 1 = xr01 (i) = q 0 1 and xr < q 0 1. Such an r exists, as otherwise i would equal q n 0 1. It can be seen that
k (i);
x x
1 + xr (i);
k (i + 1) =
0;
if k 2 [r + 1; n 0 1] if k = r if k 2 [0; r 0 1].
From this it readily follows that a(i) and a(i +1) agree in their n 0 r 0 1 leftmost positions and in their r rightmost positions. Remark 3: The above Gray code was implicitly given in [3]. In fact, it was shown in that reference that it is essentially the unique Gray code for which i) there exists a matrix A over we have that a(i) = Ax(i);
q such that for each i 2 [0; qn 0 1],
ii) for each i 2 [0; q n 0 2], the difference of the entries in the position where a(i) and a(i + 1) disagree, computed modulo q , equals 1 or q 0 1.
Let m be an integer. We describe a list of addresses that are vectors of length n over an alphabet of size q = 2m . A special instance of such an address list can be found in [2]. We shall show that this address list is a Gray code that has property P (l) for all l. We denote mn by N . For 0 i 2N 0 1, we partition the binary expansion x(i) of i (cf. Section V-B) in n binary vectors of length m = (Xn01 (i); Xn02 (i); . . . ; X0 (i)); where for each j j (i) = (xm(j +1)01 (i); xm(j +1)02 (i); . . . ; xmj (i)):
x(i) X
The ith address a(i) is defined as a (i)
and
n l q 0 vq 0 b 0 1 l q
x(i)
i
C. Gray Codes Over Alphabets of Size 2m
ug
if u is even n (vql + c); n l ugn (q 0 vq 0 c 0 1); if u is odd. We see that gn+1 (i) and gn+1 (j ) start with the same symbol, and because Gn has property P (l), they agree in the subsequent (n 0 l) posi-
n
g +1 (j )
For each vector
where the subtraction is modulo q .
;
where m = q n , and xg (i) denotes the concatenation of the symbol x and the vector g (i).
ig
B. Gray Codes Based on the q -Ary Representation of Integers
:
Consequently, gn+1 (i) and gn+1 (j ) agree in their (n + 1 0 l) leftmost positions. That is, Gn+1 has property P (l).
where An01 (i) =
= (An01 (i); An02 (i); . . . ; A0 (i))
n01 (i), and for 0 j n 0 2 j (i) = Xj (i) 8 a(j +1)m 1
X
A
where 8 denotes bitwise modulo 2 addition, and 1 denotes the all-one vector of length m.
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 12, DECEMBER 2003
Theorem 4: a(0); a(1); . . . ; a(q n 0 1) is a 2m -ary Gray code which has property P (l) for each l 2 [0; n 0 1]. Proof: Let l 2 [0; n 0 1]. Suppose that b qi c = b qj c. As b 2 i c is uniquely determined by the lm highest order coefficients in x(i), we have that Xk (i) = Xk (j ) for all k 2 [n 0 l; n 0 1], and hence Ak (i) = Ak (j ) for all j 2 [n 0 l; n 0 1]. All that is left to show is that the address list is a Gray code. To this end, let i 2 [0; 2N 0 2]. Let j be the largest index k for which Xk (i) 6= Xk (i + 1). It is clear that Ak (i) = Ak (i + 1) for k j + 1. Next, let 0 k j 0 2. As Xk (i) = Xk+1 (i) = 11 . . . 1 and Xk (i +1) = Xk+1 (i +1) = 00 . . . 0, we have that Ak (i) = Ak (i +1). Finally, we note that Xj (i) and Xj (i + 1) end in distinct symbols. As Xj01 (i) = 11 . . . 1 and Xj01 (i + 1) = 00 . . . 0, we find that Aj01 (i) = Aj01 (i + 1). Combining these results, we find that Ak (i) = Ak (i + 1) for all k
2 [0 0 1] n f g ;n
j
:
VI. RELATED PROBLEMS
B. Hamming Distance Properties of Address Lists We have considered address lists for which consecutive addresses have many common coordinates. Addresses that are close together in such lists have small Hamming distance. A problem related to ours is the following: given an integer m and an address list A = a(0); a(1); . . . ; a(M 0 1), how far must addresses be apart in the list in order that their Hamming distance equals m. In other words, one would then be interested in ; m)
fj 0 j j
= min
i
j
i; j
2 [0
;M
0 1]
; d(a(i); a(j ))
=
g
m
where d denotes the Hamming distance. Yuen [4] showed that for the binary reflected Gray code Gn (cf. Section V-A) we have that f (Gn ; m) = d 23 e. The following example shows that the binary reflected Gray code does not necessarily attain the maximum value of f (A; m). Example 2: Consider the address list
A = 0000 0001 0011 0010 0110 0100 0101 0111 ;
;
;
Proposition: Let q be even. For each n there exists a cyclic Gray code a(0); a(1); . . . ; a(q n 0 1) such that for each j 2 [0; n], a(q n 0 1) and a(q n 0 j 0 1) differ in j positions. Proof: We apply induction on n. For n = 1, the proposition is trivial. Suppose a(0); a(1); . . . ; a(q n 0 1) is a cyclic Gray code for which for each j 2 [0; n 0 1], a(q n 0 1) and a(q n 0 j 0 1) differ in j positions. By applying the reflection construction, we obtain a cyclic Gray code B = b(0); b(1); . . . ; b(q n+1 0 1). As B is cyclic, the address list b q
From a mathematical point of view, one could be interested in cyclic address lists of length M . One would then be interested in the minimal number of common coordinates of any b consecutive addresses a(i); a(i + 1); . . . ; a(i + b 0 1), where the address indexes are taken modulo M . Of course, the upper bounds from Section III are upper bounds for this number as well. The bounds are obtained by cyclic Gray codes (i.e., a(0) and a(M 0 1) differ in exactly one position) that have property P (l) for each l 2 [0; n 0 1]. For even m 2 [2; q ], the m 2 q n01 initial address of the q -ary reflected Gray code Gn from Section V-A is such a Gray code. For every integer i for which the q -ary expansion x(i) is of the form (0; 0; . . . ; 0; u; u; . . . ; u), the addresses a(0); a(1); . . . ; a(i) from the Gray code from Section V-B form such a cyclic Gray code as well. Note that such an integer i is of the form u 1 (q n0r 0 1) for some u 2 [0; q 0 1] and r 2 [0; n 0 1].
A
Our results do not imply Yuen’s result. Indeed, let b := d 23 e. Yuen’s results imply that a(i) and a(i + b 0 1) have Hamming distance at most m 0 1. As b 2 [2 + 2m02 ; 1 + 2m01 ], our results imply that a(i); a(i + 1); . . . ; a(i + b 0 1) have n 0 m common coordinates, from which we can only deduce that a(i) and a(i + b 0 1) have Hamming distance at most m. If A = a(0); a(1); . . . ; a(M 0 1) is a Gray code, then for all i and j in [0; M 0 1], a(i) and a(j ) differ in at most ji 0 j j positions. We shall show that this upper bound can be achieved.
C = ( ( n + 1)
A. Cyclic Address Lists
f(
3311
;
;
;
;
;
1111; 1110; 1100; 1101; 1001; 1011; 1010; 1000:
It is easily checked that A is a Gray code, and that f (A; m) = m for m = 1; 2; 3, while f (A; 4) = 8. According to Yuen’s result (that also is easily checked by hand), we have that f (G4 ; m) = m for m = 1; 2; 3 and f (G4 ; 4) = 6.
; b(q
n + 2); . . . ; b(qn+1 0 1); b(0); b(1); . . . ; b(qn ))
is a Gray code as well. Denoting the ith address of C by c(i), we see that c(q
while for j
2 [1
;n
n+1 0 1) = b(qn ) = 1a(qn 0 1) + 1]
n+1 0 j 0 1) = b(qn 0 j ) = 0a(qn 0 n 0 j ): c(q
From the induction hypothesis, it now follows that C satisfies the proposition. As a consequence of this proposition, there exist Gray codes of addresses of length n such that for each b, there exist b consecutive addresses that have n 0 b common coordinates. That is, the number of coordinates in which b consecutive addresses do not agree is linear in b, while Lemma 1 only yields that this number grows at least logarithmically in b. ACKNOWLEDGMENT The cooperation on this work was initiated during the meeting on the “General Theory on Information Transfer,” February 18–23, 2002, at the “Zentrum für interdisziplinäre Forschung” (ZiF) of the University of Bielefeld, Bielefeld, Germany. The first two authors wish to thank Prof. Rudolf Ahlswede for inviting them to this meeting. REFERENCES [1] M. van Dijk, S. Baggen, and L. Tolhuizen, “Coding for informed decoders,” in Proc. 2001 IEEE Int. Symp. Information Theory, Washington, D.C, June 24–29, 2001, p. 202. [2] M. W. Blüm, L. M. G. Tolhuizen, and C. P. M. Baggen, “Informed decoding in an address format,” Jap. J. Appl. Phys., pt. 1, vol. 41, no. 3B, pp. 1785–1786, Mar. 2002. [3] M. Cohn, “Affine -ary gray codes,” Inform. Contr., vol. 6, pp. 70–78, 1963. [4] K. C. Yuen, “The separability of gray codes,” IEEE Trans. Inform. Theory, vol. IT-20, p. 668, Sept. 1974.
