Bounds for Signature Codes

0.32 MB·

by  A. G. Dyachkov

Bounds for Signature Codes
Arkadii D’yachkov∗, Nikita Polyanskii†, Vladislav Shchukin∗, and Ilya Vorobyev∗
∗Lomonosov Moscow State University, Moscow, Russia
†Institute for Information Transmission Problems, Moscow, Russia
[email protected], [email protected], [email protected], [email protected]
Abstract—Wediscussupperandlowerboundsofthezeroerror We say that x(e) encodes the message e. If
capacity for signature codes based on the symmetric noiseless
x (e)(cid:44)(x (e ),...,x (e ))∈As, i∈[N], (4)
multiple access channel. i i 1 i s q
Keywords: Multiple access channel (MAC), signature is the i-th row of s-collection x(e), then the subcode (3)
7 code, symmetric MAC, compositional MAC, joinable MAC, can be written as the N-collection of rows (4), i.e., x(e) =
1 disjunctive MAC, information-theoretic bounds, random cod- {x (e),...,x (e)}.
1 N
0 ing bounds.
B. Multiple Access Channel and Signature Codes
2
I. STATEMENTOFPROBLEM We will use the terminology of a noiseless (deterministic)
b
e A. Notations multiple-access channel (MAC), which has s inputs and one
F output [1]. Let all s input alphabets of MAC be the same and
Let q, N, t, s and L be integers, q ≥ 2, N ≥ 2, 2 ≤
5 s < t, 1 ≤ L ≤ t−s, the symbol (cid:44) denotes the equality coincide with the alphabet Aq (cid:44){0,1,...,q−1}. Denote by
1 Z the finite output alphabet of size |Z|. The noiseless MAC
by definition, A (cid:44) {0,1,...,q −1} – the standard q-nary
q is prescribed by the function
] alphabet, [N] (cid:44) {1,2,...,N} – the set of integers from 1
T to N and |A| – the size of the set A. A q-nary (N×t)-matrix z =f(x ,...,x )(cid:44)f(xs), z ∈Z, xs ∈As. (5)
1 s 1 1 q
I
s. Xq =(cid:107)xi(j)(cid:107), xi(j)∈Aq, i∈[N], j ∈[t], or by the following conditional probability
c (cid:40)
[ x(j)x(cid:44)(cid:44)(x(1x(j()1,).,....,.x,xN((jt))))∈∈AANqt , (1) τ(f)(z|xs1)(cid:44) 10 iiff zz =(cid:54)=ff((xx1,,......,,xxs)) (6)
3 i i i q 1 s
5v with t columns (codewords) x(j), j ∈[t], and N rows xi, i∈ on the Cartesian product Asq×Z.
8 [N], is called a q-nary code of length N and size t=(cid:98)qRN(cid:99), Let the row xi(e), i ∈ [N], defined by (4), be the s-
0 whereafixedparameterR>0iscalledarateofthecodeXq. collection of signals at s MAC inputs at the i-th time unit.
6 Foraq-narycolumnx=(x1,...,xN)(cid:44)xN1 ∈ANq ,define Then the signal zi, zi ∈Z, i∈[N], at the output of MAC at
0 the vector of integers [N ,...,N ], where N = N (x), the i-th time unit is
0 q−1 a a
.
01 0in≤wNhiach≤xN,=a∈a.AOqb,viisouthsely,nu(cid:80)mqb−er1oNf po=sitiNon.sTi,hei∈ve[cNto]r, zi =zi(f)(e,Xq)(cid:44)f(xi(e1),...,xi(es))∈Z. (7)
7 [N ,...,Ni ], is said to be a coam=0posiation 1 of the q-nary The deterministic model of MAC is called an f-MAC.
1 vec0tor xN =q−(1x ,...,x )∈AN or, briefly, On the base of the code Xq (1) and N signals
: 1 1 N q
iv comp(cid:0)xN(cid:1)(cid:44)[N ,...,N ]. (2) z(f)(e,Xq)(cid:44)(z1(f)(e,Xq),...,zN(f)(e,Xq))∈ZN,
X 1 0 q−1
which are known at the output of MAC, an observer makes
r Note that the number of all compositions is equal
a to (cid:0)N+q−1(cid:1) and the number of all distinct q-nary columns the brute force decision about the unknown message e. To
q−1 identify e, a code Xq (1) is assigned
x = (x ,...,x ) ∈ AN having the same composition (2)
1 N q Definition 1. [2], [3]. A q-nary code X is said to be a
is equal to N!/N !.N !. Code X is said to be a fixed- q
0 q−1 q signature (s,q)-code, of size t and length N for the f–MAC
composition code if all codewords x(j), j ∈ [t], have the
if all z(f)(e,X ), e∈E(s,t) are distinct.
same composition [N ,...,N ]. q
0 q−1 Thesignaturecodeallowstosolveanidentificationproblem
Let e (cid:44) {e ,...,e }, 1 ≤ e < ··· < e ≤ t, be an
1 s 1 s of active users, arising in some communication nets. For
arbitrary s-subset of [t]. Introduce E(s,t) as the set of all
such subsets. Note that the cardinality |E(s,t)|=(cid:0)t(cid:1). For the instance, more detailed descriptions of the problem can be
s found in [3], [4].
given s-subset e={e ,...,e } called a message, consider a
1 s Let t(f)(s,q,N) be the maximal size of signature (s,q)-
non-ordered s-collection of codewords (subcode)
codesoflengthN forthef–MAC.Forfixeds≥2andq ≥2,
x(e)(cid:44){x(e ),...,x(e )}. (3) define the number
1 s
log t(f)(s,q,N)
1Inthewell-knownbook[1],theauthorsusethetermtype. R(f)(s,q)(cid:44) lim q , (8)
N→∞ N
called a rate of signature (s,q)-codes for the f–MAC. Using E. Erasure MAC
the terminology of the Shannon coding theory, the number The q-nary, q ≥ 2, symmetric f–MAC is said to be the
R(f)(s,q) can be called a zero error capacity of signature erasure MAC (briefly, eras–MAC) if it has the (q + 1)-
codes for the f-MAC. nary output alphabet Z (cid:44) {0,1,...,q−1,∗} and the output
Definition 2. [2], [5], [6]. An f–MAC given by (5) is function z =f(xs) (5) has the form:
1
said to be the symmetric f–MAC if any of s! permutations
(cid:40)
π =π(k), k ∈[s], on the set [s], satisfies the equality z =f(x ,...,x )(cid:44) a if x1 =···=xs =a, a∈Aq,
1 s
∗ otherwise.
(cid:0) (cid:1)
f(x ,...,x )=f x ,...,x ,
1 s π(1) π(s) The eras-MAC model can be considered as an adequate
π(k)∈[s], k ∈[s], π(k)(cid:54)=π(k(cid:48)), k (cid:54)=k(cid:48). (9) description for the transmission of q-nary symbols based on
the frequency modulation method.
In other words, the equality (9) means that the f-MAC is the
F. Disjunctive MAC
symmetricf-MACifthefunctionz =f(xs)doesnotdepend
1
on the order of arguments (x ,...,x ). Such symmetric f–MAC (briefly, disj–MAC) has the bi-
1 s
nary (q = 2) input and output alphabets Z (cid:44) A = {0,1}
InSect.I-C-I-F,weintroducefourmodelsofthesymmetric 2
and
f-MAC which, by our opinion, can be considered as the most (cid:40)1 if (cid:80)s x >0,
important for applications. f(x ,...,x )(cid:44) i=1 i
1 s 0 if (cid:80)s x =0.
i=1 i
C. Compositional MAC The disj-MAC model is interpreted as the transmission of
binary symbols based on the impulse modulation method.
The symmetric f–MAC is said to be the compositional In addition, the binary signature (s,2)-codes for the disj-
MAC (briefly, comp–MAC) if MAC are closely connected with the combinatorial search
theory [12] and the information-theoretic model called the
z =f(xs1)(cid:44) comp(xs1). (10) design of screening experiments [6].
Theoutlineofourpaperisasfollows.Sect.IIremindsinthe
wherethe compositionalfunction comp(xs1) isdefined by(2). form of Propositions 1-3 the principally known information-
One can easily see that the size of output alphabet for the theoreticresultsrelativetoupperandlowerboundsontherate
comp–MAC is |Z|=(cid:0)q+s−1(cid:1). Using the permutation symbol R(f)(s,q) of signature (s,q)-codes for the general case of the
s
π = π(k) (9) the necessary and sufficient condition for the symmetric f–MAC.
coincidence of signals comp(xs1) and comp(y1s) at the output InSect.III-AandIII-B,weremindthebestknown[6],[13],
of the comp–MAC can be written in the form: [14] bounds on the rate R(disj)(s,2) of signature (s,2)-codes
for the disjunctive MAC and bounds on the rate R(eras)(s,q)
comp(xs)=comp(ys) ⇐⇒ of signature (s,q)-codes for the erasure MAC.
1 1
(cid:34) s (cid:35) Theorem 1 proved in Sect. III-C gives a new combinatorial
⇐⇒ (cid:91) (cid:92) (cid:0)x =y (cid:1) (cid:54)=∅, (11) upper bound on the rate R(f)(s,q) of signature (s,q)-codes
k π(k)
π k=1 for any symmetric f–MAC.
In Sect. III-D, we study the asymptotic bounds on the rate
where the right-hand side of (11) says that for a vector xs1, R(comp)(s,q)ofsignature(s,q)-codesforthethecomp–MAC
xs1 ∈ Asq, there exists a permutation π = π(k), k ∈ [s], with large values of the parameters s and q and prove that the
remaining the comp(xs1). bound of Theorem 1 is approximately twice better than the
classical entropy bound (15) of Proposition 1. Up to now, the
D. Joinable MAC possibility to improve the entropy bound (15) for the comp-
MAC was established for the case s = q = 2 only (see, ref.
The given symmetric f–MAC (briefly, join–MAC) is de- in[6]).Inaddition,inSect.III-DweproveTheorem2yielding
scribed by the function a random coding lower bound on the rate R(comp)(s,q). If
s,q → ∞, then the comparison of upper and lower bounds
s
z =f(x ,...,x )(cid:44) (cid:91) x ⊆A . of Theorems 1 and 2, leads to Corollary 1 which claims that
1 s i q R(comp)(s,q)∼1/2.
i=1
The aim of Sect. IV is to discuss the concept of q-nary
We would like to note the paper [7], where the significant list-decoding signature codes for the joinable MAC. Such
applicationsofthecomp–MAC,calledtheB-channel,andthe codes were introduced in the recent paper [15] as a further
join–MAC, called the A-channel, were firstly developed. We developmentoftheconceptofbinarylist-decodingdisjunctive
also refer to [8]- [11], where the maximal output entropy [1], codes [13]. A combinatorial upper bound of Theorem 4
[7] of the A-channel and the B-channel was investigated in obtained in Sect. IV establishes the asymptotic (q → ∞)
different asymptotic and non-asymptotic cases. precision of the random coding bound obtained in [15].
II. INFORMATION-THEORETICBOUNDS be a probability distribution on the Cartesian product As×Z.
q
A. Entropy Upper Bound on R(f)(s,q) Using the standard symbols for the conditional probabilities
of the distribution τ (16), we denote by the symbol
Let
  {τ}(f) (cid:44)(cid:110)τ : τ(f)(z|xs)=0 ⇒ τ(z|xs)=0(cid:111)− (17)
 (cid:88)  1 1
p(cid:44) p(a):p(a)>0, p(a)=1 , (12)
  the subset of probability distributions τ (16) such that the
a∈Aq conditional probability τ(z|xs)=0 if τ(f)(z|xs)=0.
be a fixed probability distribution at the alphabet A and the 1 1
q Introduce the ∪-convex information-theoretic functions of
vector ξ1s (cid:44) (ξ1,.,ξs), ξ1s ∈ Asq, is the s-collection of inde- the argument τ ∈{τ}(f):
pendent random variables having the same distribution (12),
i.e., Pr{ξk = a} (cid:44) p(a), k ∈ [s], a ∈ Aq. Introduce the H(f)(p,τ)(cid:44) (cid:88)τ(xs,z)log τ(xs1,z) ,
corresponding Shannon entropy of the output of f–MAC, i.e, 1 q s
xs1·z τ(f)(z|xs1)· (cid:81) p(xv)
H(f)(s,q)(cid:44) v=1
p (cid:44)z(cid:88)∈Z Pr{f(ξ1s)=z}·logq Pr{f(ξ11s)=z}. (13) Ik(p,τ)(cid:44)x(cid:88)s1·zτ(xs1,z)logq τ(x(cid:81)kk1|xpsk(+x1v,)z), k ∈[s]. (18)
v=1
Usingthef–MACdefinition(6),theprobabilityintheright-
From (13) and (14), it follows that the distribution
hand side (13) can be written in the form
Pr{f(ξ1s)=z}=(cid:88)τ(f)(z|xs1)· (cid:89)s p(xk) (14) τp(f) (cid:44)(cid:40)τ(f)(z|xs1)· (cid:89)s p(xk), xs1 ∈Asq, z ∈Z(cid:41)∈{τ}(f)
xs k=1 k=1
1
and the functions (18) satisfy the equalities
The following statement called the entropy upper bound on
the rate R(f)(s,q) takes place. (cid:16) (cid:17) (cid:16) (cid:17)
H(f) p,τ(f) =0, I p,τ(f) =H(f)(s,q).
Proposition1.[2]. TherateR(f)(s,q)ofsignature(s,q)- p s p p
codes for the symmetric f–MAC satisfies the inequality Put the symbol [u]+ (cid:44)max{0;u}.
max H(f)(s,q) Proposition 2. [6]. Let s≥2, q ≥2, code rate R>0 be
p
R(f)(s,q)≤ C(f)(s,q)(cid:44) p . (15) fixed,andtheentropyHp(f)(s,q)ofafixeddistributionp(12)
s
is defined by (13). If code parameters N, t→∞ such that
B. RandomCodingErrorExponentfortheSymmetricf-MAC
log t N
Fix an arbitrary symmetric f–MAC. Given a code X , a q ∼R, a ∼p(a), a∈A , s, q−const,
q N N q
message e, e ∈ E(s,t), is said to be bad for the code X ,
q
if there exists a message e(cid:48) (cid:54)= e such that z(f)(e(cid:48),X ) = then for the FC-ensemble there exists
q
z(f)(e,X ). If the unknown message e is interpreted as the
q −log P(f)(s,t,[N ,.,N ])
random vector taking equiprobable values in the set E(s,t), lim q N 0 q−1 (cid:44)
then the relative number of ”bad” messages among all (cid:0)t(cid:1)= N→∞ N
s
|E(s,t)| messages can be considered as the error probability H(f)(s,q)
(cid:44)E(f)(s,q,R,p)>0, 0<R< p , (19)
of code Xq for the brute force decoding. Let the symbol FC s
PN(f)(s,t,[N0,.,Nq−1]) denote the average error probability and for the CR-ensemble there exists
over the fixed composition ensemble (briefly, FC-ensemble)
of t independent q-nary codewords with the same compo- −log P(f)(s,t,p)
sition [N ,.,N ]. By a similar symbol P(f)(s,t,p) we lim q N (cid:44)
0 q−1 N N→∞ N
will denote the average error probability over the completely H(f)(s,q)
randomized ensemble (briefly, CR-ensemble) of q-nary codes (cid:44)E(f)(s,q,R,p)>0, 0<R< p . (20)
CR s
X =(cid:107)x (j)(cid:107) (1) with independent components x (j) having
q i i
the same distribution p (12), i.e., Pr{x (j) = a} (cid:44) p(a), For any fixed p (12), the positive monotonically decreasing
i
i∈[N], j ∈[t], a∈A . functions E(f)(s,q,R,p) and E(f)(s,q,R,p) are ∪-convex
q FC CR
Let a symmetric f–MAC is identified as the conditional functions of the parameter R>0 of the following form:
probability τ(f)(z|xs) defined by (6). To present the results
1 E(f)(s,q,R,p)(cid:44) minE(f)(s,q,R,p,k),
about the logarithmic asymptotic behavior of probabilities FC FC
k∈[s]
P(f)(s,t,[N ,.,N ]) and P(f)(s,t,p), we need the fol-
N 0 q−1 N
lowing notations [6]. Let
E(f)(s,q,R,p,k)(cid:44)
  FC
τ (cid:44)τ(xs,z) : τ(xs,z)≥0, (cid:88)τ(xs,z)=1 (16) min (cid:110)H(f)(p,τ)+[Ik(p,τ)−kR]+(cid:111), (21)
 1 1 xs,z 1  {τ}k(f)(p)
1
and III. IMPROVEMENTSOFGENERALBOUNDS
E(f)(s,q,R,p)(cid:44) minE(f)(s,q,R,p,k), A. Bounds on the Rate R(disj)(s,2) for the Disjunctive MAC
CR CR
k∈[s]
One can easily see that the capacity of signature (s,2)-
codes for the disjunctive MAC is C(disj)(s,2)=1/s and the
E(f)(s,q,R,p,k)(cid:44) maximumintheright-handsideof(15)isattainedatthedistri-
CR butionp(12)withp(0)=21/sandp(1)=1−21/s.Thesignif-
(cid:110) (cid:111)
min H(f)(p,τ)+[I (p,τ)−kR]+ . (22) icant results relative to an improvement of the corresponding
k
{τ}(f) entropybound(15),havingtheformR(disj)(s,2)≤1/s,were
obtained in [14] for s=2 and in [13] for s≥11. In addition,
The minimum in (21) is taken over the subset {τ}(f)(p) of
k we refer to the best known asymptotic (s → ∞) lower [6]
distributions {τ}(f) (17) for which the marginal probabilities and upper [13] bounds on the rate R(disj)(s,2):
on x are fixed and coincide with p(x ) (12), k ∈[s], i.e.,
k k
2ln2 4log s
(1+o(1))≤R(disj)(s,2)≤ 2 (1+o(1)).
{τ}(f)(p)(cid:44){τ : τ ∈{τ}(f); s2 s2
k
(cid:88) (cid:88) (cid:88)τ(xs,z)=p(x ), k ∈[s]}. (23) B. Bounds on the Rate R(eras)(s,q) for the Erasure MAC
1 k
xk1−1xsk+1 z If q = 2 and s → ∞, then it not difficult to establish [16]
that the capacity of signature (s,2)-codes for the erasure
The minimum in (22) is taken over the set of all distribu- MAC is C(eras)(s,2) ∼ 1/s and the maximum in the right-
tions (17). hand side of (15) is asymptotically attained at the symmetric
Remark 1. Propositions 1-2 and the properties of the distribution p (12) with p(0) ∼ ln2/s or with p(1) ∼ ln2/s.
random error exponents (19) and (20) were formulated and In addition, we refer to the best known asymptotic lower [6]
proved in the papers [2] and [6] for the particular binary case and upper [15] bounds on the rate R(eras)(s,2):
q = 2 only. In the general case q ≥ 2, we omit the proofs 2ln2 4log s
because one can check that the given results are based on the (1+o(1))≤R(eras)(s,2)≤ 2 (1+o(1)).
s2 s2
same methods developed in [2] and [6]. Here we only note
C. Combinatorial Upper Bound for the Symmetric f-MAC
that for the symmetric f-MAC, definitions (21)-(23) leads to
the inequality Theorem 1. For any symmetric f-MAC, the rate
R(f)(s,q) of signature (s,q)-codes satisfies the inequality
E(f)(s,q,R,p) ≤ E(f)(s,q,R,p).
CR FC s+1
R(f)(s,q)≤ , s≥2, q ≥2. (25)
2s
Introduce the function
Proof of Theorem 1. Fix an arbitrary q-nary (N × t)-
E(f)(s,q,R)(cid:44)maxE(f)(s,q,R,p)>0 code Xq (1). Without loss of generality we may assume that
FC FC
p N is even, i.e., N = 2k. Note that all codewords from X
q
are distinct. For the given X introduce the bipartite graph
if0<R<C(f)(s,q),whereC(f)(s,q)isdefinedintheright- q
G = G(X ) = (V ∪ V ,E) defined as follows. For each
q 1 2
hand side (15). Hence, Propositions 1 and 2 imply that the
vertex in V (as well as in V ), we put in the correspondence
number C(f)(s,q) can be considered as the Shannon capacity the unique1q-nary vector of2 length k, |V | = |V | = qk.
1 2
of signature (s,q)-codes for the symmetric f-MAC [5].
Two vertices v ∈ V and v ∈ V are connected with an
1 1 2 2
The following statement called the random coding lower
edge iff the code X contains a codeword of length N = 2k
q
bound on the rate R(f)(s,q) of signature (s,q)-codes for
which is the concatenation of two q-nary vectors of length k,
the symmetric f-MAC can be obtained as a consequence of
correspondingtov andv .Thus,weobtainthegraphG(X )
1 2 q
Proposition 2. having n = 2qk = 2qN/2 vertices and t edges, identified by
Proposition3.[6]. TherateR(f)(s,q)ofsignature(s,q)- the elements of [t]. In addition, any message e ∈ E(s,t) is
codes for the symmetric f-MAC satisfies the inequality interpreted as a non-ordered s-collection of edges.
Let X be a q-nary signature (s,q)-code for a symmetric
R(f)(s,q) ≥ R(f)(s,q), s≥2, q ≥2, q
f-MAC.WewillcheckbycontradictionthatthegraphG(X )
q
does not contain simple cycles of length ≤2s. Let there exist
where the lower bound R(f)(s,q) is
a simple cycle of the length 2(cid:96), (cid:96) ≤ s. From the cycle we
can take the set E ⊂ [t], |E | = (cid:96), of edges, which are not
1 1
E(f)(s,q,0,p,k) intersected by vertices. Let E ⊂[t], |E |=(cid:96), E ∩E =∅,
R(f)(s,q)(cid:44)max min CR = 2 2 1 2
p k∈[s] s+k−1 denote the set of all other edges of the cycle. Consider an
min (cid:8)H(f)(p,τ)+I (p,τ)(cid:9) arbitrary subset C ⊂ [t]−(E1+E2) of the size |C| = s−(cid:96)
{τ}(f)(p) k and define two messages ei (cid:44)Ei+C ∈E(s,t), i=1,2. It is
=max min k (24)
p k∈[s] s+k−1 easy to check that outputs of the symmetric f-MAC for these
messages are the same, i.e., z(f)(e ,X )=z(f)(e ,X ). This from the set A . For the given ensemble, the probability of
1 q 2 q q
contradicts to Definition 1 of signature s-code. the event (27) satisfies the inequality
It is known (e.g., see [17]) that if a graph with n=2qN/2
s (cid:18) (cid:19)
vertices does not contain simple cycles of length ≤ 2s, then Pr{“x(j) is s-bad(cid:48)(cid:48)}≤ (cid:88) t ×
the number t of its edges is 2m−1
m=1
(cid:18) (cid:19)
t≤n1+1/s =21+1/sqN(1+1/s)/2 ≤4qN(1+1/s)/2, × 2m−1 (Pr{comp(um)=comp(vm)})N, (28)
m 1 1
i.e., the rate (8) satisfies (25).
where the definition (10) along with notations (2) are used
D. Asymptotic Bounds for the Compositional MAC andthecomponentsofvectorsum, vm ∈Am areindependent
1 1 q
For the comp-MAC, the number H(comp)(s,q) defined random variables having the same uniform distribution on the
p
set A . Applying (11), one can write that for any m∈[s],
by(13)iscalledtheShannonentropyofthe(s,p)-polynomial q
distribution. The corresponding maximization problem in the
Pr{comp(um)=comp(vm)}=
right-hand side (15) was firstly solved in [18], where the 1 1
(cid:40) (cid:34)m (cid:35)(cid:41)
author proved that the maximum is attained at the uniform (cid:91) (cid:92)(cid:0) (cid:1)
Pr u =v ≤
distribution: p(a)=1/q, a∈A , i.e., i π(i)
q
π i=1
(cid:40)m (cid:41)
sC(comp)(s,q)=max H(comp)(s,q)= (cid:92)(cid:0) (cid:1) m!
p ≤m!·Pr u =v = . (29)
p i π(i) qm
= (cid:88) s!q−s log s0!···sq−1!. (26) i=1
s !···s ! q s!q−s Inequalities (28)-(29) imply that
0 q−1
{s0,...,sq−1}
Pr{“x(j) is s-bad(cid:48)(cid:48)}≤
An asymptotic behavior of the right-hand side (26) gives
Lemma 1. If s ≥ 2 is fixed and q → ∞, then the (cid:34) t2m−1 (cid:18)m!(cid:19)N(cid:35)
function sC(comp)(s,q)=s+o(1). ≤s· max · (30)
m∈[s] m!(m−1)! qm
Proof of Lemma 1.. From the well-known extremal property
and the standard random coding arguments [6] give
of the Shannon entropy (13) it follows
(cid:20)m−log m!(cid:21)
(cid:18)q+s−1(cid:19) R(comp)(s,q)≥ min q .
sC(comp)(s,q)≤log |Z|=log =s+o(1) m∈[s] 2m−1
q q s
This leads to the statement of Theorem 2.
For q ≥ s, consider the set Z, Z ⊂ Z, of all compositions
{s ,...,s } with elements s ∈ {0,1}. Note that the size From Theorems 1 and 2 it follows
0 q−1 a
|Z|=(cid:0)q(cid:1) and the formula (26) implies that the number
s Corollary 1. If s → ∞ and q → ∞, then the rate
(cid:18)q(cid:19)s! qs R(comp)(s,q) of signature (s,q)-codes for the comp-MAC
sC(comp)(s,q)> s qs logq s!, q ≥s. satisfies the asymptotic equality R(comp)(s,q)∼ 21.
Remark 2. For the comp-MAC the asymptotic behavior
Hence if q →∞, then sC(comp)(s,q)≥s+o(1).
(q ≥2isfixed,s→∞)ofupperandlowerboundsontherate
Lemma 1 shows that for the comp-MAC with large pa- R(comp)(s,q) based on Propositions 1-2 was discussed in [6]
rameters s and q, Theorem 1 improves the classical entropy for q =2 and in [20] for q ≥3.
bound (15).
IV. LISTDECODINGCODESFORJOINABLEMAC
Theorem 2. If s ≥ 2 is fixed and q → ∞, then the
rateR(comp)(s,q)ofsignature(s,q)-codesforthecomp-MAC For any s-collection x(1),...,x(s) of columns x(j)∈AN,
q
satisfies the asymptotic inequality j ∈[s], introduce its joining
s  
ql→im∞R(comp)(s,q)≥ 2s−1. (cid:10)x(j), j ∈[s](cid:11) (cid:44) (cid:91)s x1(j),..., (cid:91)s xN(j),
Proof of Theorem 2. The proof uses a development of the j=1 j=1
method which was suggested in [19] and [6] for the binary which is a column of N subsets of A . We say that a column
q
case q =2. A codeword x(j), j ∈[t], is said to be s-bad for Q = (Q ,...,Q ), Q ⊆ A , i ∈ [N], covers a column
1 N i q
a code Xq in the comp-MAC if there exist m, m ∈ [s], and x=(x1,...,xN)∈ANq if xi ∈Qi for any i∈[N].
two disjoint messages e,e(cid:48) ∈E(m,t), e∩e(cid:48) =∅, such that Definition 3. [15]. A q-ary code X (1) is said to be a
q
j ∈e and z(comp)(e,X )=z(comp)(e(cid:48),X ). (27) list-decoding (sL,q)-code of size t and length N for join-
q q
MAC if for any s-collection of codewords (x(j ),...,x(j ))
1 s
Introducetheensembleofq-naryN×tmatricesX ,withen- its joining (cid:104)x(j ),k ∈ [s](cid:105) covers not more than L−1 other
q k
tries x (j) which are chosen independently and equiprobable codewords of code X .
i q
In the case L = 1 the list-decoding (s ,q)-code (or s- denotes a projection of the codeword x(j) on the coordinates
1
frameproof code [21]) is a signature (s,q)-code for join- i, i+1, ..., i+L−1. A codeword x(j), j ∈ [t], is said
MAC.Moreover,list-decoding(s ,q)-codeprovidesasimpler to be an L-rare in X if there exists a row index i ∈ [N]
1 q
factor decoding algorithm, that picks the unknown message such that the number of codeword indexes j(cid:48), j(cid:48) ∈[t], j(cid:48) (cid:54)=j,
e ∈ E(s,t) by searching all codewords of X covered by such that the projection xi+L−1(j(cid:48)) = xi+L−1(j), does not
q i i
the output signal z(join)(e,X )=(cid:104)x(e)(cid:105). In the general case exceed L−1. Let r = r (X ) be the number of codewords
q L q
L ≥ 1, the algorithm gives a subset of [t] that contains s which are L-rare in X . For each L-rare x(j), we can choose
q
transmitted elements and not more than L−1 extra elements. anumberi∈N,aq-naryL-sequence(a ,...,a )∈AL and
1 L q
Lett(s ,q,N)bethemaximalpossiblesizeoflist-decoding anordinalnumberofthex(j)amongall≤Lcodewordsx(j(cid:48)),
L
(s ,q)-codes of length N. For fixed s≥2, L≥1 and q ≥2, j(cid:48) ∈ [t], for which xi+L−1(j(cid:48)) = xi+L−1(j) = (a ,...,a ).
L i i 1 L
define a rate of list-decoding (s ,q)-codes: Therefore, the following claim holds.
L
Lemma 2. For any code X of length N, the number
log t(s ,q,N) q
R(s ,q)(cid:44) lim q L . r (X ) of its L-rare codewords satisfies the inequality
L N→∞ N L q
In [15] the author establishes a random coding bound on r =r (X )≤NLqL. (37)
L q
the rate of list-decoding (s ,q)-codes, which improves the
L
best previously known bounds presented in [16], [22], [23].
Lemma 3. If a q-nary code X of length N has a size
q
Theorem3.[15]. 1.Foranyfixedq ≥2,s≥2andL≥1
the following lower bound holds:
L−1
(cid:88)
−log P(q(cid:48),s,L) t>NLqL n!, (38)
R(sL,q)≥R(sL,q)(cid:44)mq(cid:48)≥axq (s+Lq−1)k(q,q(cid:48)), (31) n=0
where then there exists a subset Ls ={j1,...,jL}⊂[t] of the size
|L | = L, such that the L-sequence {x(j), j ∈ L } does not
mi(cid:88)n{q,s}(cid:18)q(cid:19)(cid:18)m(cid:19)L constain codewords which are L-rare in X . In asddition, for
P(q,s,L)(cid:44) q
m q any k ∈ [L−1] the projections of x(j ) and x(j ) on the
k k+1
m=1
coordinates 1+k(s−1), 2+k(s−1),...,L+k(s−1) are
(cid:88)m (cid:18)m(cid:19)(cid:18)m−k(cid:19)s
× (−1)k , (32) the same, i.e.,
k q
k=0
k(q,q(cid:48))(cid:44)(cid:40)1, for q =q(cid:48), (33) xL1++kk((ss−−11))(jk)=x1L++kk((ss−−11))(jk+1), k ∈[L−1]. (39)
(cid:100) q(cid:48) (cid:101), otherwise.
q−1 Proof of Lemma 3. For any j ∈ [t], we try to construct a
1
2. For any fixed q ≥2, L≥1 and s→∞ sequence L(j ) = {x(j ),x(j ),...,x(j )} of L codewords
1 1 2 L
bythefollowingrules.ThefirstelementofthesequenceL(j )
L(q−1)log e 1
R(s ,q)= q (1+o(1)), s→∞. (34) isequaltox(j ).Letasequence{x(j ),x(j ),...,x(j )}ofa
L s2(log e)2 1 1 2 k
2 lengthk,1≤k ≤L,beconstructed.Ifthelastcodewordx(j )
k
3. For any fixed s≥2 and L≥1 there exists a limit isL-rareinX ,thentheprocessendswithafailure.Ifk =L
q
L and x(jL) is not L-rare in Xq, then the process successfully
ql→im∞R(sL,q)= s+L−1. (35) ends. Otherwise, for k ≤L−1, we consider L indexes from
1+k(s−1) to L+k(s−1). Since the codeword x(j ) is
k
In [15] it was also conjectured that the lower bound (35) is
not L-rare in X we can find at least L another codewords
q
precise. We prove the conjecture in
with the same projection on the coordinates from 1+k(s−1)
Theorem 4. For any s ≥ 2, L ≥ 1 and q ≥ 2 the rate
to L+k(s−1). Among them there are ≤ k −1 ≤ L−2
R(s ,q) of list-decoding (s ,q)-codes for join-MAC satisfies
L L codewords that could be already included in the sequence at
the asymptotic inequality
the previous k−1 steps. Therefore, there exists a codeword,
L which has not been used. Among all such unused codewords,
R(s ,q)≤ . (36)
L s+L−1 we uniquely choose the codeword x(jk+1) with the cyclically
smallest index j , j >j , as a next element of L(j ).
Proof of Theorem 4. Consider an arbitrary code X of length k+1 k+1 k 1
q
Let us prove, that there exists a codeword x(j ), such that
N and size t. For a convenience of the proof, we will use 1
thedescribedprocesswillsuccessfullyend,i.e.,asaresult,we
indexes j (i) of codewords (rows) which can exceed t (N),
obtain a sequence L(j ) without L-rare codewords. The only
assumingthattheindexes(coordinates)arecyclicallyordered, 1
reasonofafailureisanemergenceofanL-rarecodeword.Fix
i.e., for instance, if a row index n>N, then for any j ∈[t],
the symbol x (j) (cid:44) x (j), where n(cid:48) (cid:44) n mod N. For a an arbitrary L-rare codeword x(j). Suppose that for some j1
n n(cid:48)
codeword x(j)∈AN, j ∈[t], we say that the symbol and sequence L(j1) we constructed x(jn)=x(j), n≤L. By
q
constructionofthesequenceL(j )weknowthatthecodeword
1
xi+L−1(j)(cid:44)(x (j),...,x (j))∈AL, i∈[N], x(j ) has the cyclically smallest index j > j among all
i i i+L−1 q n n n−1
codewords, except x(j ), ..., x(j ), and coincides with the the corresponding codeword x(l ) such that the projections
1 n−2 k
codeword x(j ) on the L coordinates: of x(l ) and x(j ) on the coordinates i , i ,..., i are
n−1 k kL k1 k2 kL
the same, i.e.,
1+(n−1)(s−1), 2+(n−1)(s−1),...,
(cid:16) (cid:17)
(L−1)+(n−1)(s−1), L+(n−1)(s−1). (40) x (j ),x (j ),...,x (j ) =
ik1 kL ik2 kL ikL kL
(cid:16) (cid:17)
Hx(ejnc)e,,exthceepctoxd(ejw)o,r.d..xx((jjn−1)),iws hthicehfihrasstthcoedseawmoerdsybmebfoorles = xik1(lk),xik2(lk),...,xikL(lk) k ∈[s]. (42)
n 1 n−2
asx(jn)ontheLcoordinates(40).Thenumberofcodewords In addition, the property (42) implies that the joining
among x(j1), ..., x(jn−2), which have the same symbols as (cid:104)x(lk),k ∈[s](cid:105) covers the sequence L(j1)={x(j), j ∈Ls}.
x(jn)andx(jn−1)ontheLcoordinates(40)isfrom0ton−2. The obtained contradiction proves Lemma 4.
Therefore, for fixed codeword x(j ) there exist ≤ n − 1
n
of possible variants for x(j ). Thus, any L-rare codeword The proof of Lemma 4 is intuitively illustrated by the
n−1
x(j), uniquely chosen as the codeword x(j ) in the sequence following two examples.
n
L (j ),spoils≤(n−1)!ofstartingcodewordsx(j ).Invirtue
s 1 1
Example 1. Let L = s = 3 and N = L+s−1 = 5. Then
of condition (38) and upper bound (37) from Lemma 2, the
code size t > rL(Xq) · (cid:80)Ln=−01n!. Therefore, there exists a three q-nary codewords x(jk), x(jk)∈A5q, k ∈[3], satisfying
starting codeword x(j ), such that the sequence L(j ) will the equalities (39) can be written in the form:
1 1
be successfully constructed and can be written in the form
x(j )= (x (j ), x (j ), x (j ), x (j ), x (j )),
1 1 1 2 1 3 1 4 1 5 1
{x(j), j ∈L }, |L |=L.
s s x(j )= (y , z , x (j ), x (j ), x (j )),
2 2 2 3 1 4 1 5 1
Lemma 4. For any list-decoding (sL,q)-code Xq of the x(j3)= (y2, z2, y3, z3, x5(j1)).
length N = s+L−1, the size t of the code X is upper
q These codewords are covered by the joining of three q-nary
bounded as follows:
codewords x(l ), x(l )∈A5, k ∈[3], which are based on the
k k q
L−1 property (42) and can be written in the form:
(cid:88)
t≤(s+L−1)LqL n!. (41)
x(l )= (x (j ), x (j ), x (j ), a , a ),
n=0 1 1 1 2 1 3 1 1 2
x(l )= (y , b , b , x (j ), x (j )),
Proof of Lemma 4. Consider an arbitrary list-decoding 2 2 1 2 4 1 5 1
x(l )= (c , z , y , z , c ).
(s ,q)-code X of the length N = s + L − 1. We prove 3 1 2 3 3 2
L q
the claim of Lemma 4 by contradiction. Assume that
Example 2. Let L=4, s=2 and N =L+s−1=5. Then
t > (s + L − 1)LqL (cid:80)L−1n!. In virtue of Lemma 3, it
is sufficient to construct thne=0subset S ⊂ [t], |S| = s, such four q-nary codewords x(jk), x(jk) ∈ A5q, k ∈ [4], satisfying
the equalities (39) can be written in the form:
that the joining (cid:104)x(j),j ∈ S(cid:105) covers every codeword of
the sequence L(j ) = {x(j), j ∈ L }, L = {j ,...,j }, x(j )= (x (j ), x (j ), x (j ), x (j ), x (j )),
1 s s 1 L 1 1 1 2 1 3 1 4 1 5 1
constructed in the proof of Lemma 3. Define a sequence P x(j )= (y , x (j ), x (j ), x (j ), x (j )),
2 2 2 1 3 1 4 1 5 1
of pairs, where each pair represents the index j of codeword x(j )= (y , y , x (j ), x (j ), x (j )),
k 3 2 3 3 1 4 1 5 1
x(j ) and the coordinate i in this codeword, such that the x(j )= (y , y , y , x (j ), x (j )).
k 4 2 3 4 4 1 5 1
symbolx (j )shouldbecoveredbythejoining(cid:104)x(j),j ∈S(cid:105):
i k
These codewords are covered by the joining of two q-nary
P ={(j1,1),(j1,2),...,(j1,N),(j2,L+1+(s−1)),... codewords x(lk), x(lk)∈A5q, k ∈[2], which are based on the
property (42) and can be written in the form:
(j ,L+1+2(s−1)),...,(j ,L+1+(k−1)(s−1))...
2 k
(j ,L+k(s−1)),...,(j ,sL)}. x(l )= (x (j ), x (j ), x (j ), x (j ), a ),
k L 1 1 1 2 1 3 1 4 1 1
x(l )= (y , y , y , a x (j )).
2 2 3 4 2 5 1
Divide this sequence of pairs into s groups g , k ∈ [s], ac-
k
cordingtotheorderoftheirappearanceinthesequenceP,i.e. To complete the proof of Theorem 4, consider an arbitrary
list-decoding (s ,q)-code X of length N, N > s+L−1,
g (cid:44){(j ,i ),...,(j ,i )}, {j ,...,j }⊂L . L q
k k1 k1 kL kL k1 kL s andsizet.DivideeachcodewordofthecodeX intos+L−1
q
Firstly, note that the projection x(j ) on the coordinates i , parts of sizes
kL k1
ik2,..., ikL is (cid:22) N (cid:23) (cid:24) N (cid:25)
n , ≤n ≤ , i∈[s+L−1].
(cid:16) (cid:17) i s+L−1 i s+L−1
x (j ),x (j ),...,x (j ) =
ik1 k1 ik2 k2 ikL kL
(cid:16) (cid:17) The number of different parts is upper bounded by the sum
= x (j ),x (j ),...,x (j ) .
ik1 kL ik2 kL ikL kL q(cid:98)s+NL−1(cid:99) +q(cid:100)s+NL−1(cid:101). Replace each part of each codeword
Secondly, from the construction of the set L described in with a unique symbol from the Q-nary alphabet of the
s
the proof of Lemma 3 it follows that codeword x(jkL) is not size Q (cid:44) 2q(cid:100)s+NL−1(cid:101). It is easy to see that the obtained
L-rare. Therefore, we can find an index l , l (cid:54)∈ L , and code X is aQ-nary list-decoding (s ,Q)-code ofthe length
k k s Q L
N = s+L−1 and the size t. Thus, the inequality (41) of
Lemma 4 implies that the size
L−1
t≤(s+L−1)L(cid:88)n!2LqL(cid:100)s+NL−1(cid:101).
n=0
The obtained upper bound immediately leads to (36).
ACKNOWLEDGMENT
TheresearchissupportedinpartbytheRussianFoundation
for Basic Research under Grant No. 16-01-00440 a.
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