On Generalized Hamming Weights for Galois Ring Linear Codes.
The definition of generalized Hamming weights (GHW) for linear codes over Galois rings is discussed. The properties of GHW for Galois ring linear codes are stated. Upper and existence bounds for GHW of ZF_4 linear codes and a lower bound for GHW of the Kerdock code over Z_4 are derived. GHW of some ZF_4 linear codes are determined.
Introduction
For any code D, -(D), the support of D, is the set of positions where not all the
codewords of D are zero, and w s (D), the support weight of D, is the weight of
-(D). For an [n; k] code C and any r, where 1 - r - k, the r-th Hamming weight
is defined [7],[14] by
d r is an [n; r] subcode of Cg:
In [1],[4], and [5] different generalizations of GHW for nonlinear case were sug-
gested. In the present paper we consider the case of Galois ring (GR) linear codes.
Outline of the paper is following.
In Section 2, we discuss the definition and state some properties of GHW of GR
linear codes. In Section 3, we show that definition of GHW from [4] determines
performance of group codes in the wire tap channel of type II. In Section 4, we find
the number of different type subcodes in a Z 4 \Gammalinear code and state a connection
between their support sizes. In Section 5, we use results of Section 4 to get bounds
for GHW of Z 4 \Gammalinear codes. In Section 6, using results of sections 4 and 5, we
obtain a lower bound for GHW of the Kerdock code over Z 4 and complete weight
hierarchys of some short Z 4 \Gamma linear codes. We also show that though the minimum
Hamming weight of a GR linear code can not exceed the minimum Hamming weight
of an optimal linear codes, heigher weights of a GR linear code do can exceed
corresponding weights of an optimal linear code of the same length and dimension.
2. Definition and Basic Properties
Let R be a Galois ring, i.e., a finite commutative ring with identity e, whose set
of zero divisors has the form pR for a certain prime p. After Nechaev's paper
Galois rings have become easy to understand for a reader with a standard
background in finite fields. In particular, using the definition given, one can prove
[10] that and the characteristic of R
(the order of e in the group (R; +)) equals p m . Since fixing the numbers p m and q m
identifies R up to isomorphism, it may be also denoted as GR(q m
of R form the following chain:
and jR quotient ring is a Galois field of order q
The ring R is constructed as a degree s Galois extension of Z p m in much the same
way as one constructs the finite field from Z p . Note also that Galois rings encompass
finite fields and residue rings as boundary cases. Namely, GF (p s
Let C be a GR linear code of length n. Let D be any GR linear subcode of C.
The support of D is defined exactly as for codes over a Galois field, i.e., the support
of D is the set of not-all-zero symbol positions of D.
In [1], the following definition was proposed for the r-th generalized Hamming
weight of C
d r linear subcode of C; log q
One can see that this definition generalizes the definition of GHW for linear codes
over Galois fields.
L. A. Bassalygo proposed another definition for support and GHW [4]. Let A be
any code of length n over arbitrary alphabet and let B be any subcode of A. Define
the support of B as follows
an such that a i 6= b i g:
In other words, the support of B is the set of positions where not all the codewords
of B have the same symbols. Define the following function
is subcode of A and
We will consider only those points where the function FA changes its values, that
is, the following two sets
are defined as follows
and 1. It is clear that the function FA (M) is completely defined by
the set of pairs so on. These pairs are called generalized
Hamming weights.
Example: Consider the nonlinear code
This code has the following values
and the function FA (M) can be depicted as follows (the function FA (M) is defined
only for integer values of M , and we draw it everywhere just for visualization)
6 FA (M)
One can see that Bassalygo's definition works for both linear and nonlinear cases.
In the case of a linear code over GF (q), we have M i =M therefore we
can study only values ffi i . In the general case, we must consider pairs
In Corollary 1, we show that if C is a linear code over GR(q
i.e., the function FC (M) can change its values only at
see that in the GR linear case there is one-to-one
correspondence between the set fd and the sets and
knowledge of values d i allows us to find values
vice versa.
Note that for GHW of nonlinear codes one more definition, nonequivalent to
Bassalygo's one, was given in [5].
Theorem 1 Let C be a linear code of length n over GR(q
and if d r
Proof: Let D be a GR linear subcode of C such that w s
log q Dg. From linearity
of D it follows that 0g.
Then log q jD
d r\Gammat (C) ! d r (C) for some t - m.
To see that there exist a code A such that d r
one can consider the code with generator matrix [1;
Note that if C is a linear code of length n over GR(q
code, then jCjjC mn. In the sequel, we suppose
that ng is the set of positions of a code of length n.
Denote by \Phi the direct sum of two sets.
ng and C
I \Phi fx Ig.
I rg.
Proof: Let S I
I g, where x \Delta y denotes the
inner product of x and y. Then log q jS I j log q jS ?
I I be such that
I I is a linear subcode of C, we see
that d r (C) - jS I j -
I rg.
Let us establish inequality in the other direction. Let D be a linear subcode of
C such that w s
I rg.
Theorem 2 Let C be a GR linear code and let C ? be its dual code. Then fd r
times
times
times
Proof: It suffices to prove that if
then there are no more than m\Gammal values d t+1
that d t (C) ! d t+1 (C); d t+m\Gammal (C) ! d t+m\Gammal+1 (C), and d t+1
for some t.
At first we show that d t (C) -
ng n -(D)
I \Phi D. Hence, jC ?
I
1, we obtain d t (C) - jI
log
Next we show that if d
a subcode of C such that log q
ng n -(D) and Hence jC I
By Lemma 1, we obtain d
where
contradiction.
Corollary 2 Let g. Then
ng
l log q M i =M
there is not ffi j such that ffi ?
Good nonlinear codes can be constructed by mapping GR linear codes into codes
over GF (q) [10],[6]. Therefore it is natural to ask whether GR linear codes themselves
can have better characteristics than linear codes over Galois fields. The
following theorem shows that GR linear codes can not be better (from the point of
view of length, cardinality, and minimum Hamming distance) than optimal linear
codes, i.e. than linear codes having the largest cardinality for a given length and
distance.
Theorem 3 Let C be a GR linear code over GR(q
Hamming distance d. Then there exist a linear code D over GF (q m ) of length n
with minimum distance d such that jDj - jCj.
Proof: We consider the case of a code C over GR(q The case of an arbitrary
m can be proved by the same way. A generator matrix of C can be written in the
are r 1 \Theta n and r 2 \Theta n GR(q
C be a subcode of C with the generator matrix
Obviously, the code b
C has the same length and the same minimum distance as
C. Note that b
C contains only elements from the ideal pGR(q us define
exterior multiplication of an element ff 2 pGR(q
by an element a 2 GR(q
Analogously,
aff n ). Now taking into account (1),
it is easy to see that b
C is a vector space over GR(q
of dimension r Hence b
C is isomorphic to some [n; r
over GF (q). Let D be a code with a generator matrix of the code b
considered
as a matrix over GF (q 2 ). Then D is a linear code over GF (q 2 ) of length n with
minimum distance d and
Using the Nordstrom-Robinson code over Z 4 as an example, we shall show later
that a GR linear code can have the same length, cardinality, and minimum distance
as an optimal linear code, and the function F for such a GR linear code can exceed
the function F for the optimal linear code.
3. An Application to The Wire-Tap Channel of Type II
One of motivations for the introduction of a notion of GHW was cryptographical
one. Ozarow and Wyner [11] considered using linear codes on the wire-tap channel
of Type II. One of their schemes uses an [n; k] linear code, say C, over GF (q). The
code has q n\Gammak cosets, each representing a q-ary (n \Gamma k)-tuple. If the sender wants
to q\Gammaary symbols of information to the receiver, he selects a random
vector in the corresponding coset. The adversary has full knowledge of the code,
but not of the random selection of a vector in a coset.
Wei showed [14] that in the linear case if the adversary is allowed to tap s symbols
(of his choice) from the sender, he will obtain r q\Gammaary symbols of information, if
and only if s - d r (C ?
It is obvious that we can use a group code on the said channel. Let C be a group
code of length n over an additive group \Theta; '. The code C has ' n =jCj cosets.
As in the linear case, the number of a coset is transmitted information and the
sender transmits a random vector from the corresponding coset. Let C have values
let the code C be used on the
wire-tap channel of type II. We prove the following theorem.
Theorem 4 Let s symbols be taped from transmitter and
the number of cosets of C that can (equiprobably) contain the transmitted vector
(with known s taped symbols) is equal to or greater than ' n\Gammas
Proof: Let I be a set of taped positions. By L denote a set of vectors that contain
zeros on positions from I , i.e., Ig. Cosets of C that
contain vectors from L form a subgroup, say \Psi, of the quotient group of cosets of
C. Cosets of C that contain vectors with given values on positions from I form a
coset of \Psi. Hence w.l.o.g. we can assume that taped symbols are zeros. Denote
by D a set of codewords of C that belong to L. It is clear that any coset of C that
belongs to \Psi has exactly jDj vectors from L and hence
The best strategy for the adversary is to choose the set I such that the cardinality
of D would be maximal. We claim that jDj - M i . Indeed, suppose that there
exists a subcode B ' C such that jBj ? M i and
the definition of GHW the support weight of any subcode of cardinality larger than
M i is equal to or greater than ffi i+1 . A contradiction. Hence the transmitted vector
equiprobably belongs to one of j\Psij cosets of C, and j\Psij - ' n\Gammas
4. Subcodes of a Z 4 \Gammalinear code
Let C be a Z 4 \Gammalinear code of length n and cardinality 4 r1 2 r2 with a generator
matrix
are r 1 \Theta n and r 2 \Theta n Z 4 \Gammamatrices. We will say that C is an
C denote a subcode of C with the generator matrix [2G 1 ] and
by e
C denote a subcode of C with the generator matrix
Remark. Throughout the rest of the paper we use symbols - and - in this meaning

Note that b
C and e
C are [n; 0; r 1 ] and [n; 0; r
an [n; s of C and let
be a generator
matrix of D, where G 1 and G 2 are s 1 \Theta n and s 2 \Theta n Z 4 \Gammamatrices respectively. By
denote the set of subcodes [n; s of C. In
the sequel, we need the following propositions.
Proposition 1
Proof: Consider the number of ways of choosing s 1 Z 4 \Gammalinearly independent
codewords. Note that linear independence over Z 4 implies that none of these codewords
consists only of zero divisors of Z 4 . The total number of codewords of C
of them consist of zero divisors of Z 4 . Hence the first
codeword, say u 1 , can be chosen by 2 distinct ways. The second
codeword u 2 can not be equal to au 1
C . Indeed, if
then either 2u are
not Z 4 \Gammalinearly independent. Thus u 2 can be chosen by 2
ways. Analogously, if we have codewords then a codeword u t can
be chosen by 2 distinct ways. Thus the number of ways of
choosing s 1 Z 4 \Gammalinearly independent codewords equals
Now we have to choose s 2 Z 2 \Gammalinearly independent codewords that consist from
zero divisors of Z 4 . These codewords should also be Z 2 \Gammalinearly independent from
the s 1 codewords chosen earlier. The number of ways for choosing these codewords
equals
Using the same arguments, one can see that the number of distinct generator matrices
of any [n; s
To complete the proof, we multiply (2) by (3) and divide by (4).
In the linear case, any nonzero codeword of a linear code belongs to one and the
same number of subcodes of a given dimension. The situation is different in the
Z 4 \Gammalinear case. We should consider the following three cases.
Proposition 2 Let u 2 C; 2u 6= 0. Then u belongs to
subcodes from -(C; s
Proof: Recall that the generator matrix of C has the form
. To prove
the proposition, we assume that one of rows of the matrix G 1 equals u and use
arguments from the proof of Proposition 1.
Proposition 3 Let
subcodes from -(C; s
Proof: Suppose a codeword u 2 D; then either
D or u 2 e
D.
Consider the case u 2 b
D. The subcode D contains 2
w such that w 2 b
D. The code C contains 2 codewords v such that
C. Hence the number of subcodes D such that u 2 b
D equals the total number
of [n; s subcodes of C multiplied by the factor 2
Using Proposition 1, we
get
Consider the case u 2 e
D. From (5) it follows that the number of subcodes D
such that u 62 b
Assuming that one of rows of G 2 equals u and using arguments like ones from
Proposition 1, we get the number of subcodes D such that u 2 e
D. This number
equals
Summation of (5) and (6) completes the proof.
Proposition 4 Let u 2 e
C. Then u belongs to
subcodes from -(C; s
Proof: Suppose that u 2 D. From the conditions of the proposition it follows
that u 62 b
D. Hence the matrix G 1 can be formed by any s 1 Z 4 \Gammalinearly independent
codewords of C. Assuming that one of rows of the matrix G 2 equals u and using
arguments from the proof of Proposition 1, we complete the proof.
By wtL (u) denote the Lee weight of a vector u.
Proposition 5 ([3], [13]) For an [n; r
Using an approach suggested in [8], we can obtain a connection between support
weight of C and supports weights of its subcodes.
Proposition 6
C)
C)
are numbers from Propositions 2,3, and 4.
Proof: Using Proposition 5 and Propositions 2,3, and 4, we get (an abbreviation
s.t. means such
u2b Cn0
u2e Cnb C
u2b Cn0
u2e Cnb C
u2b Cn0
u2e Cnb C
wtL (u)C A
u2b Cn0
u2e Cnb C
wtL (u)C A
u2b Cn0
C)
It is known that if C is a Z 4 \Gammalinear code and C   is its binary image under the Gray
map, then minimum Lee distance of C equals minimum Hamming distance of C   [6].
Such good nonlinear binary codes as Kerdock, Delsarte-Goethals, and Preparata
codes can be constructed by Gray mapping Z 4 \Gammalinear codes [6]. Therefore it is
natural to ask what is the minimal length, N(r of an [n; r
code with minimum Lee distance dL . We confine ourselves to the case of [n;
codes and obtain an analog of the Griesmer bound for N(r; 0; dL ).
Theorem 5
dL
Proof: Let C be an [n; \Gammalinear code with minimum Lee distance dL . Note
that in this case b
Using Proposition 6, we
get
Obviously, w s ( b
Assuming w s ( b
From (7) and (8) it follows that
If 3ag is an [n;
\Upsilon . Otherwise wtL (2a) ! dL . Substituting the value for N(1; 0; dL )
in (9) completes the proof.
Note that the Nordstrom-Robinson code over Z 4 meets this bound.
5. Bounds on Higher Weights
We start with Plotkin type bound.
Let d s1 ;s 2
is an [n; s subcode of Cg.
Theorem 6 If C is an [n; r
d 2s (C) - n
Proof: Calculate in two ways the sum, say S, of codeword Lee weights of all
subcodes of C.
The total number of subcodes equals -(r According to Proposition
5 the sum of Lee weights of codewords of any [n; s subcode is greater than or
equal to 2 2s1+s2 d 2s1 +s2 (C). Hence
Let A be a 2 2r1+r2 \Theta n matrix whose rows are codewords of C. W.l.o.g. we can
consider the first column of A. Any a 2 Z 4 occurs 2 2r1+r2 \Gamma2 times in the column.
belongs to A 1 [n; s
of C and there are 2 \Delta 2 2r1+r2 \Gamma2 such vectors. If a = 2 then there are three different
cases.
C. Then v belongs to A 1 [n; s subcodes of C and there are
C. Then v belongs to A 2 [n; s
subcodes of C and there are 2 r1 \Gamma1 such vectors.
C. Then v belongs to
subcofdes of C and there are 2 r1 +r2 Hence
After computations we get
Setting respectively, we get
the assertion.
Let D be a Z 4 \Gammalinear code and C be its [n; r subcode such that w s
d r1 ;r 2
(D). Let fl r1 ;r 2
and fl r1 be upper bounds for support sizes of subcodes e
C and
C.
Remark. If we don't have an information on the structure of subcodes e
D and
D then we can say that
(D). Sometimes, if we know weight
distribution of support sizes of all subcodes of codes e
D and b
D, we can find exact
values for support sizes of subcodes e
C and b
C. (see next chapter).
Now we can get a Griesmer type bound.
Theorem 7 If 2 r1 +r2 \Gammas 1
d r1 ;r 2 (D) -
otherwise
d r1 ;r 2
d r1 ;r 2
(D)
Proof: From proposition 6 we have
(D)
C)
C)
C)
C)
It is easy to check that A 2 \GammaA 3 is always nonnegative when s
Replacing w s ( b
C) by fl r1 and w s ( e
C) by fl r1 ;r 2
(C) in the case when A
by d r1 ;r 2
(D) otherwise, we get the assertion.
As corollarys we have the following relations.
Although in general d r can be equal to d r+1 , d s1 ;s 2 +1 is always greater than d s1 ;s 2
Corollary 3
Proof: Inequality d s1 ;s 2 +1 (D) - d 0;s1 +s2+1 (D) is obvious. Assuming r
(like in Theorem 7)
D)
For linear binary codes we have [8]
s
D and b
are isomorphic to linear binary codes we have
D)
D) - d 0;s1 (D) the assertion follows.
Corollary 4
ae
oe
Proof: If r
1) and the assertion follows from assuming fl r1 ;r 2
(D) and Theorem 7.
Note that as an estimation for value d 0;t (D) we can use any lower bound for
generalized Hamming weights of binary liner codes. For example using the Griesmer
bound we get
d s1 ;0 (D);
where d(C) is minimum Hamming weight of C. Note that some codes have equalities
in (10) and (11) (see Remark 5).
Our last bound is an existence one.
Theorem 8 For any s 1 - r 1 and s there exists an [n; r
Z 4 \Gammalinear code with d s1 ;s 2
- d which satisfies
d
Proof: The number of [n; r codes that contain a given subcode [n; s
equals
d
is the number of [n; s codes with supports of size less than
or equal to d. Hence if
d
is less than the total number of [n; r
at least one of [n; r codes has a subcode [n; s greater
than or equal to d. Substitution valies for the function - gives us the assertion.
6. Determination of Weight Hierarchy
It is often that we know more about subcodes e
C and b
C, wich are isomorphic to
linear binary codes. In those cases we can get better bounds.
Consider the Kerdock code over Z 4 [6],[10]. In our notation the Kerdock code
Km is an [2 code, m is odd. The minimum Lee distance of Km is
and the minimum Hamming distance is 2 . Moreover, in [6]
compositions of all codewords and weight distribution of Km were found. Suppose
l t Km such that -(u) 6=
has one of the following compositions
l
l
be the number of codewords of weight i. Then
In [6], it was also shown that subcode b
Km is isomorphic to the first order Reed-Muller
code RM(1;m) of length 2 m . To find a lower bound for d r (Km ) we need
the following lemma.
Lemma 2 The support size of any r dimensional subcode, say C, of RM(1;m)
equals to either
Proof: If C contains the all-one codeword then w s that C does
not contain the all-one codeword.
It is known [14](Theorem 5) that d r
any r rows
(that are not all-one) in a standard generator matrix generate a subcode whose
support size equals d r (RM(1;m)). Any codeword of RM(1;m) can be consedered
as a boolean linear function of m variables, say v
affine transformation of v belongs to the automorphism group of RM(1;m)
and hence it does not change the support size of C. Let f 1
functions corresponding to basis vectors of C. Assume
that f 1 (v) has a term v 1 . Then the affine transform v 1
to the form v 1 and f 2 (v) to some f 0
(v). Since f 2 (v) is linearly
independent of f 1 (v) and f 1 (v)
2 (v) has a term v j
The affine transform v 2
2 (v) to the form v 2 . Continuing
this procedure, we get f 1 that is r rows from a standard
generator matrix.
Theorem 9
d
d r
d
we have (like in Theorem 7)
where C is an [n; r of Km such that w s
C and e
C are [n; 0; r 1 ] and [n; 0; r subcodes of Km respectively, and they
are isomorphic to [n; r 1 ] and [n; r subcodes of RM(1;m). Hence from the
Lemma 2 we get
From possible compositions of codewords of Km it follows that if V is an [n; 1; 0]
subcode of Km then w s (V Hence
(3
By denote the right side of (14). It is evident that d r (Km
It is easy to check that n(r
1). On the other hand, since w s (C) - w s ( e
C), we have
linear subcode of Km ;
ae d r=2 (RM(1;m); if r is even,
d (r+1)=2 (RM(1;m); if r is odd:
It is easy to check that
d r (RM(1;m)) at r - (m \Gamma 1)=2.
It is obvious that for Km
Indeed, it is enough to take a subcode C =! a ?= f0; a; 2a; 3ag generated by
a codeword a with the composition l 0
(note that in Km there are also subcodes of cardinality
4 generated by a couple of codewords consisting of zero divisors of Z 4 ),
and check that support weight of C meets the lower bound and
The Kerdock code over Z 4 can be also defined for the case of even m [6]. In this
case
Similar to the case when m is odd we get
Theorem
d
d
Remark. Theorems 9 and 10 were obtained for the first time in [2] [3]. Later and
independently the following estimates were obtained for the Kerdock codes [13]:
d
ae P 2r
and
d
It is easy to check that estimates (16)and (17) coincides with estimates from Theorems
m=2. For the r - m=2 estimates from the theorems are
better than estimates (16) and (17).
Minimum distance of a code is only a small part of information contained in the
spectrum of distance distribution of the code. In the same way, the number of
subcodes of a given dimension with a given support weight gives much more information
on a code than only minimum support weights of subcodes. For example
we used distribution of support weights of the first order Reed-Muller code in the
proof of Theorem 9 and 10. Papers [7],[12] are devoted to studying distributions of
support weights of linear codes. We shall find weight distribution of Z 4 \Gammalinear sub-codes
of cardinality 2 3 (weight distribution of subcodes of cardinality 2 2 is evident).
Let A 3
linear subcode of Km ;
Theorem 11
A 3
A 3
A 3
Proof: We should consider subcodes
Consider an [2
generated by codewords a; 2a 6= 0; -(a) 6=
Assuming using Proposition 6, we get that
and w s
3 be sets of [2 subcodes with support weights 7
It is easy to see that if w s (! a; b
Hence,
As it was mentioned if w s any codeword a 2
C; 2a 6= 0, has Hamming weight 3 . Hence the total number of
codewords of weight 3 in all the subcodes from S 1 equals
In Km there are 2 m+1 \Gamma4 codewords b; l 0 , that are Z 4 \Gammalinearly
independent form a given a 2 Km ; 2a 6= 0. According to (18), half of these codewords
half of them form subcodes
So, a given codeword a; 2a 6= 0, belongs to 2 subcodes from S 1 . Hence the
the total number of codewords of weight 3 in all the subcodes
from S 1 equals the half number of codewords of this weight (codewords a and 3a
always belong to one and the same subcode) multiplied by 2. So, using (19)
and (12), we get
subcode of Km with support contains the codeword
Assuming s using Proposition 3, we
get that the number of an [2 subcodes with support
According to Proposition 1 the total number of an [2 subcodes equals
1). Hence the number of [2 subcodes
with support weight 7
Since 2Km is isomorphic to RM(1;m), one can see that in 2Km there are
\Theta m
subcodes of Km with support weight 2 m and
\Theta m+1\Gamma
with support weight 7
The numbers jS 1
\Theta m+1\Gamma
\Theta m; jS 3 j, and A 2
\Theta mgive the numbers of
Z 4 \Gammalinear subcodes of cardinality 8 with support weights 7
Corollary 5 The values ffi 3 and M 3 for the Kerdock code Km are
Using Theorems 2 and 7, we can find the weight hierarchy of some short Z 4 -linear
codes.
Theorem 12 The Nordstrom-Robinson code (NR) over Z 4 has weight hierarchy
Proof: The Nordstrom-Robinson code is a self-dual code with d 1
According to (15), d 2 5. From this fact and Theorem 2 it follows that
8g. The claim of the theorem
follows from this hierarchy.
Remark. The optimal linear code, say B, over GF (4) of length 8 and dimension
4 has 4: The best possible function FB (M) for such a code and the
function FNR (M) have the following form (see the picture). One can see that
5. Thus the Nordstrom-Robinson
over Z 4 code is better, in some sense, than the optimal linear code over GF (4) of
the same length and dimension. From cryptographical point of view it means that
if we use the code B on the wire tap channel the adversary must tap 4 symbols
from transmitter to get 2 bits of information, whereas in the case of using the
Nordstrom-Robinson code the adversary must tap at least 5 symbols to get 2 bits
of information. Whether there exist other GR linear codes (or codes over another
ring or additive group) that are better (from the point of view of GHW) than
optimal linear codes over GF (q) is an open question.
Theorem 13 The Kerdock code of length 16 over Z 4 has weight hierarchy
Proof: From (15) and Corollary 5 it follows that fd 1 (K 4
5 be the Preparata code over Z 4 of length 16. Using the
6 F(M)
4\GammaThe Nordstrom-Robinson code over Z 4
3\GammaThe optimal [8; 4] linear code over GF (4)
MacWilliams identities for Z 4 \Gammalinear codes [6], we see that in P 5 there exists a
codeword a such that 2a 6= 0 and 4. Using this fact and taking into
account that minimum Hamming weight of P 4 equals 4, we get d 1
Now using Theorem 2, we get the hierarchy of K 4 .
Remark. Note that if we start with known values d 1 (Km ) and d 2 (Km
then in (10) and (11) we have equality for all d s1 ;0 and d s1 ;1 .
Using Theorems 2 and 13, we can also find weight hierarchy of the Preparata
code P 4
Theorem 14 The Kerdock code of length 32 over Z 4 has weight hierarchy
Proof: From (15) and Theorem 7 it follows that fd 1 (K 5
5 be the Preparata code over Z 4 of length 32. Using
the MacWilliams identities for Z 4 \Gammalinear codes [6], we see that in P 5 there exists a
codeword a such that 2a 6= 0 and 5. Using this fact and taking into account
that minimum Hamming weight of P 5 equals 4, we get d 1
Now using Theorem 2, we get fd 5 (K 5
From Theorem 6 it follows that d 4 (K 5 or 28. Suppose d 4 (K 5
contradiction. Thus d 4 (K 5 27. The claim of the theorem follows
from the hierarchy of the values d i (K 5 ).
Using Theorems 9 and 2, we can also find weight hierarchy of the Preparata code



--R

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--CTR
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generalized Hamming weights;galois ring linear codes