Explicit Constructions of Quasi-Uniform Codes from Groups

0.11 MB·English

by  Eldho K. Thomas

Explicit Constructions of Quasi-Uniform Codes
from Groups
Eldho K. Thomas and Fre´de´rique Oggier
Division of Mathematical Sciences
School of Physical and Mathematical Sciences
Nanyang Technological University
Singapore
Email:[email protected],[email protected]
Abstract—We address the question of constructing explicitly X ,...,X be a set of quasi-uniform random variables with
1 n
3 quasi-uniform codes from groups. We determine the size of the probabilities Pr(X = x ) = 1/|λ(X )| for all A ⊆ N.
A A A
1 codebook, thealphabet and theminimumdistanceas a function
The correspondingquasi-uniformcode C of lengthn is given
0 ofthecorrespondinggroup,bothforabelianandsomenonabelian
by C = λ(X ) = {x = Pr(X = x ) > 0}.
2 groups. Potentials applications comprise the design of almost N N N N
affine codes and non-linear network codes. Quasi-uniform codes were defined in [2], where some of
n
their properties were discussed, and importantly their weight
Ja I. INTRODUCTION enumerator polynomial was computed.
7 Let X1,...,Xn be a collection of n jointly distributed For a linear (n,k) code C of dimension k and length n
2 discrete random variables over some alphabet of size N. We (over some finite field), the weight enumeratorpolynomialof
denote by A a subset of indices from N = {1,...,n}, C is defined as

] and X = {X , i ∈ A}. We call the support of X
R A i A W (x,y)= xn−wt(c)ywt(c),
λ(X )={x :Pr(X =x )>0}. C
G A A A A cX∈C
. Definition1. Aprobabilitydistributionoverasetofnrandom
h wherewt(c)istheweightofc,thatisthenumberofnon-zero
variables X ,...,X is said to be quasi-uniform if for any
t 1 n coefficients of c. For arbitrary codes, rather than the weight
a A⊆N, X is uniformly distributed overits supportλ(X ):
m A A of the codewords, the distance between two codewords is of
[ P(XA =xA)=(cid:26) 01/|λ(XA)| iofthxeArw∈isλe.(XA), binetetrheestd.iLsteatncAerp(cro)fi=le|o{fcC′ ∈ceCn,ter|e{dj a∈t cN. N, octje6=thact′jw}|e=avor}id|

1 defining A using wt(c−c′), since this already assumes that
The motivation for introducingquasi-uniformrandom vari- r
v
the difference of two codewords makes sense. It was shown
ables [1] is that they possess a non-asymptotic equipartition
8
in[2]thatquasi-uniformcodesaredistance-invariant,meaning
2 property, by analogy to the asymptotic equipartition property,
3 where long typical sequences have total probability close to that the distance profile does not depend on the choice of c.
6 1, and are approximately uniformly distributed. Theorem 1. [2] Let C be a quasi-uniform code of length n.
.
1 We call a code C of length n an arbitrary nonemptysubset Then its weight enumerator W (x,y) = n A xn−jyj is
0 ofX1×···×Xn whereXi isthealphabetfortheithcodeword given by C Pj=0 j
3
symbol, and each X might be different.
:1 We can associateito every code C a set of random vari- WC(x,y)= qH(XN)−H(XA)(x−y)|A|yn−|A|, (1)
v ables [2] by treating each codeword (X ,...,X ) ∈ C as a AX⊆N
1 n
i
X random vector with probability where H(X ) = log (|λ(X )|) is the joint entropy of the
A q A
r 1/|C| if x ∈C, induced codeword symbol quasi-uniform random variables.
a P(X =x )= N
N N (cid:26) 0 otherwise.
The formula for the weight enumerator shows that it only
To the ith codeword symbol then corresponds a codeword depends H(X ). In fact, [2] which introduced quasi-uniform
A
symbol random variable X induced by C. codesfocusedon theirinformationtheoreticproperties,rather
i
than on their coding properties. The goal of this paper is
Definition 2. [2] A code C is said to be quasi-uniform if
to address the construction and understanding of such quasi-
the induced codeword symbol random variables are quasi-
uniform codes from a constructivepoint of view. We will use
uniform.
thegrouptheoreticapproachproposedin[3],[4]forconstruct-
Givena code,we explainedabovehowto associate a setof ing quasi-uniform random variables from finite groups.
random variables, which might or not end up being quasi- More precisely, in Section II, we recall the construction
uniform. Conversely, given a set of quasi-uniform random of quasi-uniform codes from groups, and compute the size
variables, a quasi-uniform code is obtained as follows. Let of the corresponding code as a function of the group G we
started with. We then consider abelian groups in Section III, {g ,g h ,...,g h },byassumingwlogthath istheidentity
1 1 2 1 m 1
and computethe alphabetas well as the minimumdistanceas element of G . In words, we observe that every element in
N
afunctionofG.WenextmovetononabeliangroupsinSection g G is a multiple of a non-trivial element of G . Thus,
1 N N
IV.Thestructureofgroup,eventhoughitisanonabelianone, when computing the above table, we have (by reordering the
allows in some cases to mimic a definition for the minimum elements of G so as to list first the elements in g G ):
1 N
distance of the code. Potential applications to the design of
G ... G
1 n
almost affine codes is mentioned in Section V.
g g G g G
Lookingatconstructionsofcodesfromgroupsismotivated 1 1 1 1 n
g h g h G =g G g h G =g G
bytheneedtodesignnon-linearcodesfornetworkcoding(see 1 2 1 2 1 1 1 1 2 n 1 n
. . .
. . .
[5]forapplicationsofquasi-uniformcodestonetworkcoding), . . .
apartfrom designingalmostaffine codesas mentionedabove. g h g h G =g G ... g h G =g G
1 m 1 m 1 1 1 1 m n 1 n
. . .
. . .
II. QUASI-UNIFORM CODES FROM GROUPS . . .
Let G be a finite group of order |G| with n subgroups where g h G = g G , for all i, j = 1,...,n because by
1 i j 1 j
G1,...,Gn, and GA =∩i∈AGi. Given a subgroup Gi of G, definition of GN, hi ∈ Gj. Since the cosets of GN partition
the(left)cosetofGi inGisdefinedbygGi ={gh, h∈Gi}. G, we will get |G|/|GN copies of a code C.
The number of (left) cosets of Gi in G is called the index of One of the motivations to consider quasi-uniform codes is
Gi inGandisdenotedby[G:Gi].ItisknownfromLagrange that they allow to go beyond abelian structures. Nevertheless,
Theoremthat[G:Gi]=|G|/|Gi|.IfGi isnormal,thesetsof we willstartbyconsideringthe case ofabeliangroups,which
cosets G/Gi :={gGi, g ∈G} are themselves groups, called is easier to handle.
quotient groups.
Let X be a random variable uniformly distributed over G, III. QUASI-UNIFORM CODES FROM ABELIANGROUPS
that is P(X = g) = 1/|G|, for any g ∈ G. Define the new Suppose that G is an abelian group, with subgroups
random variable Xi =XGi, with support the [G:Gi] cosets G1,...,Gn. The procedure from Section II explains how to
of Gi in G. Then P(Xi = gGi) = |Gi|/|G| and P(Xi = obtainaquasi-uniformdistributionofnrandomvariables,and
gGi, i∈A)=|∩i∈AGi|/|G|. thusaquasi-uniformcodeoflengthnfromG.Toavoidgetting
This shows that quasi-uniform random variables may be several copies of the same code, as exposed in Lemma 1,
obtained from finite groups. More precisely: notice that since G is abelian, all the subgroups G ,...,G
1 n
are normal, and thus so is G . If |G | > 1, we consider
Theorem2. [3],[4]ForanyfinitegroupGandanysubgroups N N
instead of G the quotient group G/G , and we can thus
G ,...,G of G, there exist n jointly distributed quasi- N
1 n
assume wlog that |G |=1.
uniform discrete random variables X ,...,X such that for N
1 n
all non-empty subsets A of N, Pr(XA =xA)=|GA|/|G|. Lemma 2. The size of the code alphabet is ni=1[G:Gi].
Quasi-uniformcodesareobtainedfromthesequasi-uniform Proof:ItisenoughtoshowthatallthecPosetsthatappear
distributionsbytakingthesupportλ(XN),asexplainedinthe in the table are distinct. By definition, every column contains
introduction. Codewords (of length n) can then be described [G : G ] distinct cosets. If |G | 6= |G |, the respective cosets
i i j
explicitlybylettingtherandomvariableX takeeverypossible will have different sizes, so let us assume that |G | = |G |.
i j
values in the group G, and by computing the corresponding If gG = g′G , then g−1gG = G = g−1g′G and it must
i j i i j
cosets as follows: be that g−1g′G is a subgroup.Thisimplies that g−1g′ ∈G ,
j j
G ... G and thus that g−1g′G =G .
1 n j j
g g G g G The size of the alphabetcan oftenbe reduced,as explained
1 1 1 1 n
g2 g2G1 g2Gn next. Let πi denote the canonical projection πi :G→G/Gi.
. . . Since G/G is itself an abelian group,let us denote explicitly
. . . i
. . .
by ψ : G/G → H this group isomorphism. Then π (g) =
i i i i
g g G ... g G
|G| |G| 1 |G| n gG 7→h∈H via ψ (gG )=h, i=1,...,n.
i i i i
Each row corresponds to one codeword of length n. The
Proposition 1. Let G be an abelian group with subgroups
cardinality |C| of the code obtained seems to be |G|, but in
G ,...,G . Then its corresponding quasi-uniform code is
fact,itdependsonthesubgroupsG ,...,G .Indeed,itcould 1 n
1 n defined over H ×···×H .
bethattheabovetableyieldsseveralcopiesofthesamecode. 1 n
Proof: Let X be again this random variable defined over
Lemma 1. Let C be a quasi-uniform code obtained from a
G by Pr(X = g) = 1/|G|. Define a new random variable
group G and subgroups G ,...,G . Then |C| = |G|/|G |.
1 n N Z by Z = ψ (π (X)) which takes values directly in H .
In particular, if |G |=1, then |C|=|G|. i i i i i
N Then Pr(Z = h) = Pr(ψ (π (X)) = h) = Pr(π (X) =
i i i i
Proof: Let G = {h ,...,h } be the intersection of gG )=|G |/|G|. Similarly, if Pr(Z =h ,i ∈A)> 0, then
N 1 m i i i i
all the subgroups G ,...,G . There are |G|/|G | cosets Pr(Z = h : i ∈ A) = Pr(ψ (π (X)) = h : i ∈ A) =
1 n N i i i i i
of G in G. Let us compute a first coset, say g G = Pr(π (X)=gG :i∈A)=|G |/|G|.
N 1 N i i A
In other words, we get a labeling of the cosets which We thus get a code of length n = p + 1, containing p2
respects the group structures componentwise. The next result codewords, which is linear over F (by using that C is iso-
p p
then follows naturally. morphictotheintegersmodp).Sincethepairwiseintersection
of subgroups is trivial, the minimum distance is p.
Corollary 1. A quasi-uniform code C obtained from an
We finish this section by providing a worked out example.
abelian group is itself an abelian group.
Example 1. Consider the elementary abelian group G =
Proof: First notice that the zero codeword is in
C × C ≃ {0,1,2} × {0,1,2} and the four subgroups
C, since the codeword corresponding to the identity ele- 3 3
G = h(1,0)i = {(0,0),(1,0),(2,0)}, G = h(0,1)i =
ment in G is (ψ (π (G )),...,ψ (π (G ))) = (0,...,0) 1 2
1 1 1 n n n {(0,0),(0,1),(0,2)}, G = h(1,1)i = {(0,0),(1,1),(2,2)},
where each 0 corresponds to the identity element in each 3
and G =h(1,2)i={(0,0),(1,2),(2,1)}. Using the method
abelian group H . Let (ψ (π (gG )),...,ψ (π (gG ))) and 4
i 1 1 1 n n n ofSection II, we obtainthefollowingcodewords(we write ij
(ψ (π (g′G )),...,ψ (π (g′G ))) be two codewords in C.
1 1 1 n n n instead of (i,j) for brevity):
ThennotethatthecodewordinCcorrespondingtotheelement
g+g′ ∈G is (ψ (π ((g+g′)G )),...,ψ (π ((g+g′)G )))
1 1 1 n n n
where ψ (π (g+g′))=ψ (π (g))+ψ (π (g′)), i=1,...,n. h(10)i h(01)i h(11)i h(12)i
i i i i i i (00) h(10)i h(01)i h(11)i h(12)i
Every codeword has an additive inverse for the same reason.
(01) (01)(11)(21) h(01)i (01)(12)(20) (01)(10)(22)
It forms an abelian group because every group law compo- (02) (02)(12)(22) h(01)i (10)(21)(02) (02)(11)(20)
nentwise is commutative. (10) h(10)i (10)(11)(12) (10)(21)(02) (01)(10)(22)
Because every H is an abelian group, we can freely use 0 (11) (01)(11)(21) (10)(11)(12) h(11)i (02)(11)(20)
i
(12) (02)(12)(22) (10)(11)(12) (01)(12)(20) h(12)i
since it corresponds to the identity element of H , as well as
i (20) h(10)i (20)(21)(22) (01)(12)(20) (02)(11)(20)
the operation + and −, since + is the group law for H , and
i (21) (01)(11)(21) (20)(21)(22) (10)(21)(02) h(12)i
−istheadditiveinverse.However,thealphabetHi ispossibly (22) (02)(12)(22) (20)(21)(22) h(11)i (01)(10)(22)
anyabeliangroup,inparticular,differentabeliangroupsmight
Now let H = G/h(10)i, H = G/h(01)i, H = G/h(11)i
be used for different components of the codewords. The 1 2 3
and H = G/h(12)i. Note that H ≃ C = {0,1,2} for all
classification of abelian groups tells us that each H can be 4 i 3
i i. If we replace the subgroupsby their quotients in the above
expressed as the direct sum of cyclic subgroups of order a
table, we get the following code from Lemma 1:
prime power. In the particular case where we have only one
cyclic group, then (1) the group Cpr is isomorphic to the h(10)i h(01)i h(11)i h(12)i
integers mod pr, and (2) the group C is isomorphic to the (00) 0 0 0 0
p
integersmodp,whichinfacthasafieldstructure,andwedeal (01) 1 0 1 1
with the usualfinite field F . If all the subgroupsG ,...,G (02) 2 0 2 2
p 1 n
have index p, then we get an (n,k) linear code over F . (10) 0 1 2 1
p
The minimum distance of an abelian quasi-uniformcode is (11) 1 1 0 2
encoded in its weight enumerator, but is however not easily (12) 2 1 1 0
read from (1). We can easily express it in terms of the (20) 0 2 1 2
subgroups G ,...,G . (21) 1 2 2 0
1 n
(22) 2 2 0 1
Lemma 3. The minimum distance min wt(c) of a quasi-
c∈C
uniform code C generated by an abelian group G and its It is a ternary linear code of length p+1 = 4 and minimum
subgroups G , i=1,...n is n−max |A|. distance p=3 with generator matrix
i A∈N,GA6={0}
Proof: The minimum distance minc6=c′∈C|{ci 6= c′i}| 1 0 1 1 .
can be written as minc6=c′∈Cwt(c−c′) since c−c′ makes (cid:18) 0 1 2 1 (cid:19)
sense. Furthermore, since c−c′ ∈C, it reduces, as for linear
codes over finite fields, to minc∈Cwt(c), and we are left to Since H(XN)=logq9, H(XA)=logq3 when |A| =1 and
find the weight of the codeword having maximum number of H(XA)=logq9if|A|≥2,wehavethatqH(XN)−H(XA) =9
zeros. This corresponds to finding the maximum number of when A is empty, qH(XN)−H(XA) = qlogq(9/3) = 3 when
subgroups whose intersection contains a non-trivial element |A|=1 and 1 otherwise, so that the weight enumeratorof C
of G. is, using Theorem 1:
As anillustration,hereisa simplefamilyof abeliangroups
4
that generate (n,k) linear codes over Fp. WC(x,y) = 9y4+ 3(x−y)y3
(cid:18)1(cid:19)
Lemma 4. The elementary abelian group Cp×Cp generates 4 4
a (p+1,2) linear code over F with minimum distance p. + (x−y)2y2+ (x−y)3y+(x−y)4
p (cid:18)2(cid:19) (cid:18)3(cid:19)
Proof:ThegroupG=Cp×Cp containsp+1non-trivial = x4+8xy3,
subgroups, of the form h(1,i)i where i = 0,1,...p−1 and
h(0,1)i.Theyallhaveindexpandtrivialpairwiseintersection. as is clearly the case.
IV. QUASI-UNIFORM CODES FROM NONABELIANGROUPS
C ×C ×C
3 2 2
Suppose now that G is a nonabelian group. The resulting D12 (cid:17)
(cid:17)
quasi-uniform codes might end up being very different de- (cid:0) 6(cid:17)
pending on the nature of the subgroups considered. We next 6(cid:0) 2 (cid:17) 2
(cid:0) (cid:17)
treat the different possible cases that can occur.
hr3i hr2,si 1×1×C2 C3×C2×1
A. The Case of Quotient Groups QQ
IfGisanonabeliangroup,butG ,...,G arenormalsub- @@2 6 QQ2 6
1 n @ Q
groups,thentheintersectionG ofallsubgroupsG ,...,G Q
N 1 n
1
is a normal subgroup. Following Lemma 1, we can further 1
consider the quotient G/G . As a result, some nonabelian
N
groups are really reduced to abelian ones. This is the case of Fig. 1. On the right, the dihedral group D12, and on the left, the abelian
some dihedral groups. Let groupC3×C2×C2,bothwithsomeoftheirsubgroups.
D =hr,s | rm =s2 =1, rs=sr−1i
2m
Corollary2. Theminimumdistancemin wt(c)ofaquasi-
c∈C
be the dihedral group of order 2m.
uniform code C generated by a nonabelian group G and its
Lemma 5. Quasi-uniform codes obtained from dihedral normalsubgroupsG ,i=1,...nisn−max |A|,
i A∈N,GA6={0}
groupswhose order is a power of2 and some (possiby all) of where the weight wt(c) is understood as the number of
their normal subgroups are obtained from abelian groups. components which are not an identity element in some H .
i
Proof: Normal subgroups H of D are known: H is
2m Example 2. The dihedral group D has two normal sub-
either a subgroup of hri, or 2|m and H is one of the two 12
groups G = hr3i and G = hr2,si with trivial intersec-
maximal subgroups of index 2 hr2,si, hr2,rsi. Since m is 1 2
tion (see Figure 1). We can create quasi-uniform codes of
a power of 2, the intersection G of (any choice of) these
N length n by choosing n−2 other normal subgroups. Since
normal subgroups is necessarily a subgroup of order some
D /G ≃D , this gives a nonabelian quotient H .
powerof 2, and we can take the quotient D /G , which is 12 1 6 1
2m N
a dihedralgroup of smaller order also a powerof 2. But then Thisexampleillustrates thedifferencebetweenan informa-
its normal subgroups will be of the same form, and the same tion theoretic view of these codes, where one focuses on the
process can be iterated, until we reach D which is abelian. joint entropy of the corresponding quasi-uniform codes, and
4
a coding perspective, where the actual code, its alphabet, and
The case when G is nonabelian and all the subgroups its structure are of interest. We observe on Figure 1 that if
G ,...,G arenormalbutsome(possiblyall) ofthequotient we care about having two (normal) subgroups G ,G with
1 n 1 2
groups G/G are nonabelian is very interesting. Indeed, in respective order 2 and 6, in a group of order 12, we could
i
that case, Proposition 1 still holds (with H nonabelian), and have done that with the abelian group C ×C ×C . The
i 3 2 2
infact,thecorrespondingquasi-uniformcodestillhasagroup difference is in the alphabet: D /G is a nonabelian group,
12 1
structure, but that of an nonabelian group. This gives the while C ×C ×C /1×1×C is abelian. This question of
3 2 2 2
opportunityto gobeyondabelianstructures,andyettokeep a distinguishing nonabelian groups whose entropic vectors can
group structure. As recalled above, we may assume wlog that orcannotbeobtainedfromabeliangroupswasaddressedmore
|G |=1. generally in [6].
N
Lemma 6. A quasi-uniform code C obtained from a non- Definition 3. [6] Let G be a nonabelian group and let
abelian group G with normal subgroups G ,...,G where G ,...,G be fixed subgroups of G. Suppose there exists
1 n 1 n
at least one quotient group G/G is nonabelian forms a an abelian group A with subgroups A ,...,A such that for
i 1 n
nonabelian group. every non-empty A⊆N, [G:G ]=[A:A ]. Then we say
A A
that (A,A ,...,A ) represents (G,G ,...,G ).
The proof is identicalto that of Corollary 1, but one has to 1 n 1 n
be cautiousthat the grouplaw is notcommutative.The group It follows immediately that if (A,A ,...,A ) represents
1 n
law ∗ is defined componentwise, and the identity element is (G,G ,...,G ), then the quasi-uniform codes generated by
1 n
the codeword (1 ,...,1 ). both groups have the same joint entropy, and in turn the
H1 Hn
It is then possible to mimic the definition of minimum dis- same weight enumerator, by Theorem 1. This is the case for
tance.Theweightofacodewordisthenthenumberofcompo- dihedral and quasi-dihedral 2-groups as well as dicyclic 2-
nentswhicharenotanidentityelement.Theidentitycodeword groups,whichareabeliangrouprepresentableforanynumber
playsthe roleofthe wholezerocodeword.Theminimumdis- of subgroups, and all nilpotent groups for n=2 [6].
tanceminc6=c′∈C|{ci 6=c′i}|isthenminc6=c′∈Cwt(c∗(c′)−1). If (G,G1,...,Gn) cannot be abelian represented, then
Indeed,(c′)−1 =((c′)−1,...,(c′ )−1),andnoncommutativity we can build a quasi-uniform code using exactly the same
1 n
is not an issue, since every inverse componentwise is both a subgroupsG ,...,G and be sure that its weightenumerator
1 n
left and a right inverse.This gives a counterpartto Lemma 3. cannot be obtained from an abelian group. It was also shown
in [6] that dihedral groups whose order is a power of 2 are almost affine codes are quasi-uniform. It is thus natural to
abelian grouprepresentable,howeverLemma 5 is stronger,in look for such codes among codes built from groups. Because
that it shows that the code alphabet will be the same. of the definition of almost affine codes, p-groups are the first
candidates that come to mind.
B. The Case of Nonnormal Subgroups
Lemma 7. Let G be a p-group, and let G ,...,G be
Let G be a nonabelian group, with subgroups G ,...,G , 1 n
1 n
subgroups of index p. The corresponding quasi-uniform code
where some(possiblyall) of the subgroupsare notnormal.In
is almost affine.
that case, we lose the group structure on the set of cosets. To
start with, Lemma 1 will still hold, since it does not depend Proof: First note that G ,...,G are normal subgroups
1 n
on the normality of the subgroups, however in general, we of G, thus so is their intersection, and wlog we may assume
cannot take the quotient by GN anymore, and the copy of C that|GN|=1.ThenProposition1holds(thoughtheHimight
we are left with does not have a group structure. This is an be nonabelian), and we obtain a p-ary quasi-uniform code.
exampleofsituationsmentionedintheintroduction,wherethe Since any intersection G will have order a power of p, the
A
minimum distance is then not of interest. code obtained is almost affine.
There are other ways to get almost affine codes from p-
Example 3. Consider the group S =
3
groups, and p-groups are not the only finite groups that can
{(),(12),(13),(23),(123),(132)} of permutations on
provide almost affine codes.
three elementsdescribedincycle notation.Thecorresponding
quasi-uniform code is: VI. CONCLUSION
h(12)i h(13)i h(23)i Quasi-uniform codes were known to be constructed from
() h(12)i h(13)i h(23)i groups. In this paper, we were interested in relating the prop-
(12) h(12)i (12),(132) (12),(123) erties of the obtained code as a functionof the corresponding
(13) (13),(123) h(13)i (13),(132) group. We determined the size of the code, its alphabet, and
(23) (23),(132) (23),(123) h(23)i its minimum distance, both for abelian groups, but also for
(123) (13),(123) (23),(123) (12),(123) some nonabelian groups where the group structure allows to
(132) (23),(132) (12),(132) (13),(132) mimic the definition of minimum distance. An application to
the design of almost affine codes is also given.
Note that we could label the cosets using integers, however
Currentandfutureworksinvolvestudyingfurtherproperties
then one should keep in mind that these integers do not have
ofthesecodescomingfromgroups,andinparticular(1)codes
any algebraic meaning.
coming from nonabelian groups which cannot be reduced to
Losing the group structure of the code however has the abelian groups, and (2) almost affine codes. The information
advantage that we have more flexibility in choosing the theoretic point of view is also of course of interest: it is
subgroupswe deal with, and thus have more choices in terms related to the understanding of entropic vectors coming from
of possible intersections that we are getting. nonabeliangroups.Alongtermgoalisthedesignofnon-linear
network codes.
Proposition 2. There exist a quasi-uniform code obtained
from a non-nilpotent group which cannot be obtained by any ACKNOWLEDGMENT
abelian group.
The work of E. Thomas and F. Oggier is supported by
Proof: Let G be a non-nilpotent group. Then G is not the Nanyang Technological University under Research Grant
abelian group representable for all n [6]. That is, there exists M58110049.
some subgroups G such that [G: G ] 6= [A: A ] for some
i A A REFERENCES
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