Table Of ContentMacWilliams Identity for the Rank Metric
Maximilien Gadouleau and Zhiyuan Yan
Department of Electrical and Computer Engineering
Lehigh University, PA 18015, USA
E-mails: {magc, yan}@lehigh.edu
Abstract—Thispaperinvestigatestherelationshipbetweenthe The rest of the paper is organized as follows. Section II
rankweightdistributionofalinearcodeandthatofitsdualcode. reviewsnecessarybackgroundsinanefforttomakethispaper
Themainresultofthispaperisthat,similartotheMacWilliams
self-contained. Section III-A introduces the concepts of q-
7 identityfortheHammingmetric,therankweightdistributionof
0 any linear code can be expressed as an analytical expression product and q-derivative for homogeneous polynomials, and
0 of that of its dual code. Remarkably, our new identity has investigates their properties. Using these tools, Sections III-B
2 a similar form to the MacWilliams identity for the Hamming and III-Cprovethe MacWilliams identity for the rank metric,
n metric. Our new identity provides a significant analytical tool and Section III-D derives the relationship between the mo-
a to the rank weight distribution analysis of linear codes. We mentsofthe rankdistributionof alinearcodeandthoseofits
J use a linear space based approach in the proof for our new
dualcode.Wealsoprovideanalternativederivationoftherank
identity,andadaptthisapproachtoprovideanalternativeproof
6 of the MacWilliams identity for the Hamming metric. Finally, distribution of MRD codes in Section III-E. Some examples
1
we determine the relationship between moments of the rank are provided in Section III-F to illustrate our results. Finally,
distribution of a linear code and those of its dual code, and Section IV adapts the approach in Sections III-B and III-C to
]
T provideanalternative derivation of therankweight distribution provide an alternative proof of the MacWilliams identity for
of maximum rank distance codes.
I the Hammingmetric.All the proofshave beenomitted dueto
.
s limited space, and they will be presented at the conference.
c I. INTRODUCTION
[
The rank metric has attracted some attention due to its II. PRELIMINARIES
1
relevance to wireless communications [1], [2], public-key
v A. Rank metric
cryptosystems [3], and storage equipments (see, for example,
7
9 [4]).Duetotheseapplications,thereisasteadystreamofwork Consider an n-dimensional vector x =
0 that focuson generalpropertiesof codeswith the rank metric (x0,x1,...,xn−1)∈GF(qm)n. Assume {α0,α1,...,αm−1}
1 [4]–[14]. Despite these works, many open problems remain is a basis set of GF(qm) over GF(q), then for
70 for rank metric codes. For example, it is unknown how to j =0,1,...,n−1, xj can be written as xj = mi=−01xi,jαi,
0 derive the rank weight distribution for any given linear code where xi,j ∈ GF(q) for i = 0,1,...,mP− 1. Hence,
/ except when the code is a maximum rank distance (MRD) xj can be expanded to an m-dimensional column vector
cs code[5].Besidestheminimumrankdistance,therankweight (x0,j,x1,j,...,xm−1,j)T with respect to the basis set
: distribution is an importantproperty of any rank metric code, {α0,α1,...,αm−1}. Let X be the m×n matrix obtained by
v
and determines its error performance in applications. expanding all the coordinates of x. That is,
i
X In this paper, we investigate the rank weight properties of
r linear codes. The main result of this paper is that, similar to x0,0 x0,1 ... x0,n−1
a x1,0 x1,1 ... x1,n−1
twheeigMhtacdWisitlrliibaumtisonideonftiatynyfolrintehaer Hcoadmemcianng bmeeterxicp,rethsseedranaks X= ... ... ... ... ,
an analytical expression of that of its dual code. Our new xm−1,0 xm−1,1 ... xm−1,n−1
identity is a significant analytical tool for both rank weight
distribution and hence error performance analysis of linear where xj = mi=−01xi,jαi. The rank norm of the vector
codes. To our best knowledge, no similar result exists in the x (over GF(q)P), denoted as rk(x|GF(q)), is defined to be
def
literature. It is also remarkable that our new MacWilliams the rank of the matrix X over GF(q), i.e., rk(x|GF(q)) =
identity for the rank metric has a similar form to that for rank(X) [5]. In this paper, all the ranks are over the base
the Hamming metric. Despite the similarity, our new identity field GF(q) unless otherwise specified. To simplify notations,
is proved using a different approach based on linear spaces. we denote the rank norm of x as rk(x) henceforth.
Using the same approach, we give an alternative proof of the The rank norm of x is also the number of coordinates in
MacWilliams identity for the Hamming metric. Based on our x that are linearly independent over GF(q) [5]. The field
newidentity,wealsoderiveanexpressionthatrelatesmoments GF(qm) may be viewed as an m-dimensional vector space
of the rank distribution of a linear code to those of its dual overGF(q). Thecoordinatesof x thusspana linearsubspace
code,andprovideanalternativederivationfortherankweight of GF(qm), denoted as S(x), and the rank of x is the
distribution of MRD codes. dimension of S(x).
Forallx,y∈GF(qm)n,itiseasilyverifiedthatd (x,y)d=ef Gaussian binomial [16], defined as n = α(n,u)/α(u,u).
R u
rk(x−y) is a metric over GF(qm)n, referred to as the rank Notethat n isthenumberofu-dime(cid:2)ns(cid:3)ionallinearsubspaces
u
metric henceforth [5]. The minimum rank distance of a code, of GF(q)(cid:2)n.(cid:3)We also define β(m,0) d=ef 1 and β(m,u) d=ef
dpeonssoitbeldeapsaidrsR,oifsdsiismtipnlcyt cthoedemwionridmsu.m rank distance over all ui=−01 m1−i foru>0,whichareusedinSectionIII-A.These
Qtermsa(cid:2)recl(cid:3)oselyrelatedtotheGaussianbinomial:β(m,u)=
m β(u,u) and β(m+u,m+u)= m+u β(m,m)β(u,u).
B. The Singleton bound and MRD codes u u
(cid:2) (cid:3) (cid:2) (cid:3)
The minimum rank distance of a code can be specifically
III. MACWILLIAMS IDENTITYFOR THERANK METRIC
bounded. First, the minimum rank distance d of a code
R
of length n over GF(qm) is obviously bounded above by A. q-product and q-derivative of homogeneous polynomials
min{m,n}.CodesthatsatisfydR =marestudiedin[8].Also, Definition 4 (q-product): Let a(x,y;m) =
it can be shown that dR ≤ dH [5], where dH is the minimum ri=0ai(m)yixr−i and b(x,y;m) = sj=0bj(m)yjxs−j be
Hamming distance of the same code. Due to the Singleton Ptwo homogeneouspolynomialsin x anPd y of degrees r and s
bound on the minimum Hamming distance of block codes respectively with coefficients a (m) and b (m) respectively.
i j
[15], the minimum rank distance of a block code of length n a (m) and b (m) for i,j ≥ 0 in turn are real functions of
i j
(n≤m) and cardinality M over GF(qm) thus satisfies m, and are assumed to be zero unless otherwise specified.
The q-product of a(x,y;m) and b(x,y;m) is defined to be
d ≤n−log M +1. (1)
R qm def
the homogeneous polynomial of degree (r+s) c(x,y;m) =
Asin[5],werefertocodesthatachievetheequalityinEq.(1) a(x,y;m)∗b(x,y;m)= r+s c (m)yuxr+s−u, with
u=0 u
as MRD codes. It is also shown that the dual of any MRD
P
code is also an MRD code [5]. Clearly MRD codes are the u
counterparts of maximum distance separable (MDS) codes. cu(m)= qisai(m)bu−i(m−i).
Xi=0
C. Weight enumerator and Hadamard transform
For n≥0 the n-th q-power of a(x,y;m) is defined recur-
We restrict our attention to the Hamming metric and the sively:a(x,y;m)[0] =1anda(x,y;m)[n] =a(x,y;m)[n−1]∗
rank metric only henceforth in this paper.
a(x,y;m) for n≥1.
Definition 1 (Weight function): Let d be a metric over
Toillustratetheq-product,weprovidesomeexamplesofthe
GF(qm)n, and define w(v) = d(0,v) as a weight over q-product.Wehavex∗y =yx,y∗x=qyx,yx∗x=qyx2,and
GF(qm)n. Suppose v ∈ GF(qm)n has weight r, then the yx∗(qm−1)y =(qm−q)y2x.Notethatx∗y 6=y∗x.Itiseasy
weight function of v is defined as f (v)=yrxn−r.
w to verify that the q-product is in general non-commutative.
We shall henceforth denote the Hamming weight function
However, it is commutative for some special cases.
and the rank weight function as f and f respectively. Note
H R Lemma 1: Suppose a(x,y;m) = a is a constant indepen-
that n is the maximum weight for both f and f .
H R dent from m, then a(x,y;m) ∗ b(x,y;m) = b(x,y;m) ∗
Definition 2: Let C be a code of length n over GF(qm).
a(x,y;m) = ab(x,y;m). Also, if deg[c(x,y;m)] =
Suppose there are A codewords in C with weight i, then the
i deg[a(x,y;m)], then [a(x,y;m)+c(x,y;m)]∗b(x,y;m) =
weight enumerator of C, denoted as WC(x,y), is defined as a(x,y;m)∗b(x,y;m)+c(x,y;m)∗b(x,y;m),andb(x,y;m)∗
n [a(x,y;m)+c(x,y;m)]=b(x,y;m)∗a(x,y;m)+b(x,y;m)∗
WCw(x,y)d=ef fw(v)= Aiyixn−i. c(x,y;m).
Definition 3 (HadamavXr∈dCtransformXi[=105]): Let C be the The homogeneous polynomials al(x,y;m) d=ef [x+(qm −
field of complex numbers. Let a ∈ GF(qm) and let 1)y][l] and bl(x,y;m)d=ef(x−y)[l] turn out to be very impor-
{1,α1,...,αm−1} be a basis set of GF(qm). We thus have tant to our derivations below. The following lemma provides
a = a0 +a1α1 +...+am−1αm−1. Finally, let ζ ∈ C be the analytical expressions of al(x,y;m) and bl(x,y;m).
a primitive q-th root of unity. We define χ(a) d=ef ζa0. For Lemma 2: For i ≥ 0, σi d=ef i(i−21). For l ≥ 0, we have
a mapping f from F to C, the Hadamard transform of f, y[l] =qσlyl and x[l] =xl. Furthermore,
denoted as fˆ, is given by
l
l
fˆ(v)d=ef χ(u·v)f(u). (2) al(x,y;m)= (cid:20)u(cid:21)α(m,u)yuxl−u, (3)
uX∈F uX=0
l
l
D. Notations bl(x,y;m)= (−1)uqσuyuxl−u. (4)
(cid:20)u(cid:21)
In orderto simplify notations,we shall occasionallydenote uX=0
the vector space GF(qm)n as F. We denote the number of Note that al(x,y;m) is the rank weight enumerator of
vectors of rank u (0 ≤ u ≤ min{m,n}) in GF(qm)n as GF(qm)l.
N (qm,n). It can be shown that N (qm,n) = n α(m,u), Definition 5 (q-transform): We define the q-transform of
whuere α(m,u) is defined as follouws: α(m,0)(cid:2)u=(cid:3) 1 and a(x,y;m) = ri=0ai(m)yixr−i as the homogeneous poly-
α(m,u) = ui=−01(qm −qi) for u ≥ 1. The nu term is the nomial a¯(x,y;Pm)= ri=0ai(m)y[i]∗x[r−i].
Q (cid:2) (cid:3) P
Definition 6 (q-derivative [17]): For q ≥ 2, the q- MRD codes has been derived in [5]. However, we will use
derivative for x6=0 of a real-valued function f(x) is defined ourresults to give an alternativederivationof the rankweight
as distributionofMRD codeslater,andthuswe shallnotusethe
f(qx)−f(x)
f(1)(x)d=ef . result in [5] here.
(q−1)x Lemma 7: For r ≥ 1, suppose vr = (v0,...,vr−1) ∈
Theq-derivativeoperatorislinear. Forν ≥0,we shalldenote GF(qm)r has rank r ≤ m. Then the number of vectors in
⊥
the partial ν-th q-derivative of f(x,y) (with respect to x) as Lr =hvri with rank r, denoted as Ar,r, depends on only r
f(ν)(x,y). The 0-th q-derivative of f(x,y) is defined to be andsatisfiesAr,r =α(m,r−1)−qr−1Ar−1,r−1.Furthermore,
f(x,y) itself. the rank weight enumerator of Lr is given by
Lemma 3: For ν ≤n, the ν-th q-derivative of the function
WR (x,y)=q−m [x+(qm−1)y][r]+(qm−1)(x−y)[r] .
xn is given by β(n,ν)xn−ν. Also, the ν-th q-derivative Lr n o
of f(x,y) = r f yixr−i is given by f(ν)(x,y) = Thefollowinglemmarelatestherankweightenumeratorof
ri=νfiβ(i,ν)xiP−νi.=0 i a cLoedmemtoat8h:atLoeftaCny⊆ofGitFs(sq-mth)rorbdeeraelleinmeeanrtcaorydeexwteitnhsiroannsk.
PLemma 4 (Leibniz rule): For two homogeneouspolynomi- 0
als f(x,y) = ri=0fiyixr−i and g(x,y) = sj=0gjyjxs−j wbeeitghhetreannukmweeriagthotreWnuCRm0(exr,ayto)r, oafnditsfosr-ths o≥rd0er, Blet-eWlemCRse(nxt,ayry)
withdegreesrPandsrespectively,theν-th(ν ≥P1)q-derivative
extension C . Then WR (x,y) does not depend on B and is
of their q-product is given by s C
s
given by
ν
ν
(f(x,y)∗g(x,y))(ν) = q(ν−l)(r−l)··· WR (x,y)=WR (x,y)∗[x+(qm−1)y][s]. (8)
(cid:20)l(cid:21) Cs C0
Xl=0 Combining Corollary 1, Lemma 7, and Lemma 8, the rank
··· f(l)(x,y)∗g(ν−l)(x,y). (5) weight enumerator of hvi⊥ can be determined at last.
Next, we derive the q-derivatives of a (x,y;m) = [x + Proposition 1: For v ∈ GF(qm)n with rank r ≥ 0, the
l
(qm−1)y][l] and b (x,y;m)=(x−y)[l]. rank weight enumerator of L=hvi⊥ dependson only r, and
l
Lemma 5: For ν ≤l we have is given by
al(ν)(x,y;m) = β(l,ν)al−ν(x,y;m) (6) WLR(x,y) = q−m [x+(qm−1)y][n]+(qm−1)···
bl(ν)(x,y;m) = β(l,ν)bl−ν(x,y;m). (7) ··· (x−ny)[r]∗[x+(qm−1)y][n−r] . (9)
o
B. The dual of a vector
C. MacWilliams identity
As an important step toward our main result, we derive
the rank weight enumerator of hvi⊥, where v ∈ GF(qm)n UsingtheresultsshowninSectionIII-B,wenowderivethe
MacWilliams identity for rank metric codes.
is an arbitrary vector and hvi d=ef {av:a∈GF(qm)}. It is Lemma 9: Suppose that for all λ ∈ GF(qm)∗ and all
⊥
remarkable that the rank weight enumerator of hvi depends u ∈ F, we have w(λu) = w(u). For v ∈ GF(qm)n, let
on only the rank of v. ⊥
us denote hvi as L. Then the Hadamard transform of the
Definition 7: For s ≥ 1 the s-th order B-elementary weight function f , denoted as fˆ , satisfies
w w
extension of an (n,k) linear code C is the (n + s,k +
0
s) linear code defined as Cs d=ef {(c0,...,cn+s−1) ∈ WLw(x,y)=q−m WFw(x,y)+(qm−1)fˆw(v) . (10)
GF(qm)n+s|(c0,...,cn−1) − (cn,...,cn+s−1)B ∈ C0}, Lemma 10: Supposhe v ∈ GF(qm)n has rank r.iThen the
where B is an s × n matrix over GF(q). The 0-th order Hadamard transform of the rank weight function is given by
elementary extension of C is defined to be C itself.
Lemma 6: Let C0 be a0n (n,k) linear code0 over GF(qm) fˆR(v)=(x−y)[r]∗[x+(qm−1)y][n−r]. (11)
with generator matrix G and parity-check matrix H . The Let C be an (n,k) linear code over GF(qm), and let
ss)-thlinoerdaerrcBod-eeleCmsenotvareyr0eGxFte(nqsmio)n wofithC0aisgtehneer(anto+r 0ms,aktri+x WanCdR(xW,CyR⊥)(=x,yP) ni==0Ainjy=i0xBn−jyijbxen−itjs braentkhewreaingkhtweeniugmhtereantuo-r
G 0 merator of its dualPcode C⊥.
Gs = (cid:18) B0 Is (cid:19) and a parity-check matrix Hs = Theorem 1: For any linear code C and its dual code C⊥,
H −H BT .
0 0 1
(cid:0) Corollary 1: Su(cid:1)ppose v=(v0,...,vn−1)∈GF(qm)n has WCR⊥(x,y)= |C|W¯CR(x+(qm−1)y,x−y), (12)
⊥
rank r ≥ 1. Then L = hvi is equivalent to the (n−r)-th
order elementary extension of an (r,r−1) linear code with where W¯CR is the q-transform of WCR. Equivalently,
d =2. n n
R
It is easy to verify that the (r,r −1) code with d = 2 is B yjxn−j =qm(k−n) A (x−y)[i]∗[x+(qm−1)y][n−i].
R j i
actually an MRD code as defined in Section II-B. Xj=0 Xi=0
We hence derive the rank weight enumerator of an (r,r− (13)
1,2) MRD code. Note that the rank weight distribution of Also, Bj’s can be explicitly expressed in terms of Ai’s.
Corollary 2: We have We remark that the above rank distribution is consistent with
n that derived by Gabidulin in [5].
1
B = A P (i;m,n), (14)
j i j
|C|Xi=0 F. Examples
where In this section, we illustrate Theorem 1 and Proposition 2
j using some examples. For m ≥ 2, consider the (3,2) linear
i n−i
Pj(i;m,n)d=ef (cid:20)l(cid:21)(cid:20)j−l(cid:21)(−1)lqσlql(n−i)α(m−l,j−l). code C1 over GF(qm) with generator matrix
Xl=0
(15) 1 α 1
G = ,
1 (cid:18) 1 α 0 (cid:19)
D. Moments of the rank distribution
Next, we investigate the relationship between moments of where α is a primitive element of GF(qm). It can be verified
the rank distribution of a linear code and those of its dual thattherankweightenumeratorofC isgivenbyWR (x,y)=
1 C1
code. Our results parallel those in [15, p. 131]. x3+(qm−1)yx2+q2(qm−1)y2x+(qm−q2)(qm−1)y3.
First, applying Theorem 1 to C⊥, we obtain Applying Theorem 1, we obtain WR (x,y) = x3 +(qm −
C⊥
1
n n 1)y2x. We can verify by hand that WR (x,y) is indeed the
Aiyixn−i =qm(k−n) Bjbj(x,y;m)∗an−j(x,y;m). rank weight enumerator of C⊥, which hC1⊥as a generator matrix
Xi=0 Xj=0 H =( −α 1 0 ). ForC1, bothsides of (17) are givenby
(16) 1 1
q2m,qm 3 ,(qm−1+ 3 ),and1forν =0,1,2,3respectively.
By q-differentiating Eq. (16) ν times with respect to x and 1 1
using the Leibniz rule in Lemma 4 as well as the results in Note tha(cid:2)t t(cid:3)he results h(cid:2)o(cid:3)ld when m=2<n=3.
Lemma 5, we obtain For m ≥ 4, let us now consider the (4,2) code C2 over
Proposition 2: For 0≤ν ≤n, GF(qm) with the following generator matrix
n−ν n−i ν n−j 1 α α2 α3
(cid:20) ν (cid:21)Ai =qm(k−ν) (cid:20)n−ν(cid:21)Bj. (17) G2 =(cid:18) 1 αq α2q α3q (cid:19).
Xi=0 Xj=0
As in [15], we refer to the left hand side of Eq. (17) as C is actually a (4,2) MRD code with d = 3. Hence, its
2 R
moments of the rank distribution of C. We remark that the dual code C⊥ is also a (4,2) MRD code with d = 3.
2 R
cases where ν =0 and ν =n are trivial. Also, Proposition 2 The rank weight enumerators of both C and C⊥ can be
2 2
can be simplified if ν is less than the minimum distance of readily obtained using Proposition 3, and they are given
the dual code. by WR (x,y) = WR (x,y) = x4 + 4 (qm − 1)y3x1 +
If Cνo<rodll′a,rtyhe3n: Let d′R be the minimum rank distance of C⊥. q2m−C21− 41 (qm−C2⊥1) y4.Itcanbeve(cid:2)r1ifi(cid:3)edthatWCR2(x,y)
R (cid:8)and WCR⊥(x,(cid:2)y)(cid:3) satisfy T(cid:9)heorem 1. For C2, it can also be
n−ν n−i A =qm(k−ν) n . (18) verified2that both sides of (17) are q2m, 41 qm, 42 , 41 , and
(cid:20) ν (cid:21) i (cid:20)ν(cid:21) 1 for ν =0,1,··· ,4 respectively. (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
Xi=0
Finally, consider the (7,4) code C over GF(24) with the
E. Rank distribution of MRD codes 3
following generator matrix
The rank distribution of MRD codes was first given in [5].
BasedonourresultsinSectionIII-D,weprovideanalternative 1 0 0 0 β3 β6 β12
derivation of the rank distribution of MRD codes. In this 0 1 0 0 β6 β12 0
G = ,
subsection, we assume n≤m. 3 0 0 1 0 β12 0 β3
First, we obtain the following results necessary for our 0 0 0 1 0 β3 β6
alternative derivation of the rank distribution.
whereβ isaprimitiveelementofGF(24).Itsrankweightenu-
Lemma 11: Let {a }l and {b }l be two sequences
j j=0 i i=0 meratoris givenbyWR (x,y)=x7+105y2x5+7350y3x4+
of real numbers. Suppose that for 0 ≤ j ≤ l we have C3
a = j l−i b . Then for 0 ≤ i ≤ l we have b = 58080y4x3, Theorem 1 indicates that the rank weight enu-
j i=0 l−j i i merator of its dual code is given by WR (x,y) = x7 +
ij=0(−P1)i−j(cid:2)qσi−(cid:3)j ll−−ji aj. 465y3x4+3630y4x3, which can be verifieCd3⊥using exhaustive
PBased on Corolla(cid:2)ry 3(cid:3)and using Lemma 11, we can derive
search. It can also be verified that both sides of (17) for C
the rank distribution of MRD codes when n≤m: 3
are 216,520192,682752,196416,22416,2772,127, and 1 for
Proposition 3 (Rank distribution of MRD codes): Let C be
an (n,k,d ) MRD code over GF(qm) (n ≤ m), and let ν =0,1,··· ,7 respectively.
R
WR(x,y) = n A yixn−i be its rank weight enumerator.
C i=0 i IV. MACWILLIAMS IDENTITYFORTHEHAMMING METRIC
We then havePA0 =1 and for 0≤i≤n−dR,
In this section, we adapt the approach used in our proof of
i
n d +i
A = (−1)i−jqσi−j R qm(j+1)−1 . Theorem1toprovideanalternativeproofoftheMacWilliams
dR+i (cid:20)dR+i(cid:21)Xj=0 (cid:20)dR+j(cid:21)(cid:16) (cid:17) identityfortheHammingm⊥etric.WefirstderivetheHamming
(19) weight enumerator of hvi , where v is an arbitrary vector.
Then, using this result and properties of the Hadamard trans- We remark that the MacWilliams identities for the Hamming
form, we obtain the MacWilliams identity for the Hamming and the rank metrics given in Theorems 2 and 1 respectively
metric. have exactly the same form except for the q-transform in
Definition 8: Fors≥1,thes-thordercoordinateextension Eq. (12). Note that Theorem 2 is precisely the MacWilliams
ofan(n,k)linearcodeC isdefinedasthe(n+s,k+s)code identity for the Hamming metric given by Theorem 13 in
0
Cs d=ef{(c0,...,cn+s−1)∈GF(qm)n+s|(c0,...,cn−1)∈C0}. [15, Chap. 5], although our proof is different from that in
The 0-th order coordinate extension of C is defined as C [15, Chap. 5]. Finally, we remark that Theorem 13 in [15,
0 0
itself. Chap. 5] is a special case of the MacWilliams Theorem
Weremarkthatthes-thordercoordinateextensionisaspecial for complete weight enumerators (see Theorem 10 in [15,
case of the s-th order B-elementary extension with B=0. Chap. 5]). For the rank metric, it is not clear how we can
Lemma 12: Let C be an (n,k) linear code over GF(qm), adapt the concept of complete weight enumerator to give a
0
with a generator matrix G and a parity-check matrix proof of the MacWilliams identity.
0
H . Then C over GF(qm) has a generator matrix G =
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⊥
r ≥ 1. Then L = hvi is equivalent to the (n−r)-th order 1998.
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We hence derive the Hamming weight distribution of an
2760,Oct.2003.
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relies on the MacWilliams identity, and thus may not be used
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⊥ [5] E. M. Gabidulin, “Theory of codes with maximum rank distance,”
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WH (x,y)=q−m{[x+(qm−1)y]r+(qm−1)(x−y)r}. [7] W.B.Vasantha andN.SureshBabu,“Onthecovering radius ofrank-
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The followinglemma relates the Hammingweightenumer-
[8] K. N. Manoj and B. Sundar Rajan, “Full Rank Distance codes,”
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0 rankmetric,” Proc.InformationTheoryWorkshop,p.215,Oct.2002.
Hamming weight enumerator WH (x,y), and for s ≥ 0
C0 [10] W.B. Vasantha and R. S.Raja Durai, “Maximum rank distance codes
let WH (x,y) be the weight enumerator of its s-th order with complementary duals: an application to F-adder channel,” Dec.
C
coordinsate extension C . Then 2002.
s [11] U.SripatiandB.SundarRajan,“Ontherankdistanceofcycliccodes,”
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Cs C0 [12] A. Kshevetskiy and E. M. Gabidulin, “The new construction of rank
Combining Corollary 4, Lemma 13, and Lemma 14, the codes,” Proc. IEEEInt. Symp. onInformation Theory, pp.2105–2108,
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mined. [13] E. M. Gabidulin and P. Loidreau, “On subcodes of codes in the rank
metric,” Proc. IEEE Int. Symp. on Information Theory, pp. 121–123,
Proposition 4: For v ∈ GF(qm)n with wH(v) = r, the Sept.2005.
⊥
Hamming weight enumerator of L = hvi depends on only [14] P.Loidreau, “Properties ofcodesinrankmetric,” preprint.
[15] F.MacWilliamsandN.Sloane,TheTheoryofError-CorrectingCodes.
w (v), and is given by
H Amsterdam:North-Holland, 1977.
[16] G.E.Andrews, The Theory ofPartitions, ser. Encyclopedia ofMathe-
WLH(x,y) = q−m [x+(qm−1)y]n+(qm−1)··· matics andits Applications, G.-C.Rota, Ed. Reading, MA:Addison-
n Wesley,1976,vol.2.
··· (x−y)r[x+(qm−1)y]n−r . (21) [17] G. Gasper and M. Rahman, Basic Hypergeometric Series, 2nd ed.,
Lemma 15: Suppose v ∈ GF(qm)n has Hamoming weight ser. Encyclopedia of Mathematics and its Applications. Cambridge
University Press,2004,vol.96.
r. Then the Hadamard transform of the Hamming weight
function is given by
fˆ(v)=(x−y)r[x+(qm−1)y]n−r. (22)
H
Using Lemma 15, we finally establish the MacWilliams
identity for the Hamming metric.
Theorem 2: For any linear code C, we have
1
WH (x,y)= WH(x+(qm−1)y,x−y). (23)
C⊥ |C| C
