In this section, we will state and prove the main results about the homogeneous weights modulo pe of linear codes over Galois rings.
We introduce our main result for linear codes over GR(p`, m) of this chapter in the following theorem:
Theorem 2.11. Suppose that C is a linear code over GR(p`, m) of type
(p`m)k1(p(`−1)m)k2· · ·(pm)k`.
Suppose, also, thatNChom(j, pe)denotes the number of codewords inCthat have homogeneous weights congruent toj modulope, then
NChom(j, pe)≡0 (mod pq), j= 0,1, . . . , pe−1
where
q = max
½ 0,
¹k1+k2+· · ·+k`−pe−(`−1)m−1 (p−1)(pe−(`−1)m−1)
º¾
and e≥(`−1)m+ 1.
Proof. Before proving the theorem, we note that, the homogeneous weight of every codeword inC is divisible by p(`−2)m, and so, we can introduce a new weight function w0 by letting
w0(x) = 1
p(`−2)mwhom(x), x∈GR(p`, m). (2.7)
Then we can write this new weight as:
w0(x) :=
0 ifx= 0
pm if 06=x∈p`−1GR(p`, m) (pm−1) otherwise.
Now, suppose thatNC0 (j, pe) denotes the number of codewords inCthat have thew0-weights congruent toj modulo pe, then, we see that
NChom(j, pe) =NC0 (j/p(`−2)m, pe−(`−2)m). (2.8)
So, it is enough to prove the result for NC0 (j, pe) first and then we will use (2.8) to extend it toNChom(j, pe).
Suppose that the code C has a generating matrix of the form that appears in Theorem 2.7 above, and let
{c1, . . . , ck1, b1, . . . , bk2, . . . , a1, . . . , ak`}
be the rows of the matrix, i.e., the generators of C. So, ci’s are GR(p`, m)-independent,
and {pci, bj|i, j} are independent inpGR(p`, m) and so on and finally
©p`−1ci, p`−2bj, . . . , ak¯
¯i, j, . . . , kª
are independent inp`−1GR(p`, m). Let ˜Cbe the linear code overGR(p`, m) that is generated by
©p`−1c1, . . . , p`−1ck1, p`−2b1, . . . , p`−2bk2, . . . , a1, . . . , akmª .
Then, we note that ˜C is a linear code over p`−1GR(p`, m) and is (k1 +k2 +· · ·+k`)- dimensional by the type ofC. We also note that, ifwHdenotes the Hamming weight, then we have
w0(c) =pmwH(c), ∀c∈C.˜ (2.9) But this means that, if NCH(j, pe) denotes the number of codewords in C that have their Hamming weights congruent toj modulo pe, then we have
NC0˜(j, pe) =NCH˜(j/pm, pe−m), (2.10)
for allj= 0, pm, . . . , pe−pm. But note that applying Theorem 2.1 to ˜C with the Hamming weight gives us
νp¡
NCH˜(j/pm, pe−m)¢
≥
¹k1+k2+· · ·+k`−pe−m−1 (p−1)pe−m−1
º
for all j and e≥m+ 1, putting this into (2.10) gives us
νp¡
NC0˜(j, pe)¢
≥
¹k1+k2+· · ·+k`−pe−m−1 (p−1)pe−m−1
º
(2.11)
for all j and e≥m+ 1. Since the result in Theorem 2.1 is actually true for the cosets of linear codes as well, we see that we actually have
νp¡
Na+ ˜0 C(j, pe)¢
≥
¹k1+k2+· · ·+k`−pe−m−1 (p−1)pe−m−1
º
(2.12)
as well, wherea∈ µ
p`−1GR(p`, m)
¶n .
Now, by the choice of ˜C, we see that C can be written as the union of a finite number of cosets of ˜C. We already have the result for p`−1GR(p`, m)-cosets of ˜C. So, now let a∈GR(p`, m)n be any codeword and suppose that we are looking at the coset
A=a+ ˜C.
We will apply induction on r, the number of coordinates in a that are in GR(p`, m) \ p`−1GR(p`, m).
If r= 0, then the result is proved by (2.12).
Now, suppose the result in (2.12) is proven for all cosets that have up tor−1 coordinates inGR(p`, m)\p`−1GR(p`, m) and suppose that ahas r such coordinates. Without loss of generality we might assume thatastarts with such a coordinate, and sincew0(x+y) =pm−1 for all x∈GR(p`, m)\p`−1GR(p`, m) and y∈p`−1GR(p`, m) we can assume that astarts with 1. So we can write
A=a+ ˜C = 10 +b+ ˜C = 10 +B
whereB =b+ ˜C is a coset of ˜C withbstarting with 0 and bhavingr−1 coordinates from GR(p`, m)\p`−1GR(p`, m). Since
w0(1 +p`−1x) =pm−1
for all x∈GR(p`, m) we see that
NA0(j, pe) =NB0˙(j−pm+ 1, pe), (2.13)
for all j= 0,1, . . . , pe−1, where ˙B is B with its first coordinate deleted. But notice that B˙ = ˙b+ ˙˜C
with ˙˜C denoting, in the same way, ˜C with its first coordinate deleted. Now, we can apply the induction hypothesis to ˙B because we might still assume that ˙B is of length nby just adding a zero coordinate to it. Note that, because of the type of the generating matrix that C˜has, ˙˜Cis either still (k1+k2+· · ·+k`)-dimensional orpmcopies of a (k1+k2+· · ·+k`−1)- dimensional code. So, applying the induction hypothesis and using (2.13), we get
νp¡
NA0(j, pe)¢
≥
¹k1+k2+· · ·+k`−pe−m−1 (p−1)pe−m−1
º
or
νp¡
NA0 (j, pe)¢
≥m+
¹k1+k2+· · ·+k`−1−pe−m−1 (p−1)pe−m−1
º .
But since the latter is greater than or equal to the former whenever e≥m+ 1, and p is a prime, we see that, we get
νp¡
NA0(j, pe)¢
≥
¹k1+k2+· · ·+k`−pe−m−1 (p−1)pe−m−1
º
(2.14)
for allj and e≥m+ 1 where A is any coset of ˜C. But since the original code C is just a union of a finite number of cosets of ˜C, the result of the theorem now follows easily from (2.14) and (2.8).
We used the fact thate≥(`−1)m+1 in the latter parts of the proof of the main theorem.
Since every weight is divisible by p(`−2)m because of the structure of the homogeneous weight, we still have to figure out what happens when we have (`−2)m+ 1≤e≤(`−1)m.
For the remaining part of this section, we will let C be a linear code of the same type as in Theorem 2.11, and ˜C be the same code as was defined in the proof above. We first note that
whom(a+c)≡whom(a) (modpe) (2.15) for all c ∈C,˜ a∈¡
GR(p`, m)¢n
when (`−2)m+ 1 ≤e ≤(`−1)m. This implies that we have the following quick corollary for this case:
Corollary 2.12. With C being the same as in Theorem 2.11, we have
NChom(j, pe)≡0 (modpm(k1+k2+···+k`))
for all j, and (`−2)m+ 1≤e≤(`−1)m.
It turns out however that we have a better result than Corollary 2.12 and we can give the result in the following theorem:
Theorem 2.13. Suppose that C is a linear code over GR(p`, m) of type
(p`m)k1(p(`−1)m)k2· · ·(pm)k`.
Then, for (`−2)m+ 1≤e≤(`−1)m, we have
NChom(j, pe)≡0 (mod pq), j= 0,1, . . . , pe−1
where
q=m(k1+k2+· · ·+k`)+
¹k1+k2+· · ·+k`−1−pe−(`−2)m−1 (p−1)(pe−(`−2)m−1)
º .
Proof. We will again replacewhombyw0andNChom(j, pe) byNC0 (j, pe) and when we do that, we will have replaced e by e−(`−2)m and so we will assume that 1≤ e≤m. Suppose C has the same generators as in the proof of Theorem 2.11, and let ˜C be the same code as was defined in the proof of the theorem. We know that
C= [
a∈S
(a+ ˜C) (2.16)
whereS is the set that is defined as
S=
½Xk1
i=1
αici+
k2
X
j=1
βjbj+· · ·+
kX`−1
t=1
γtdt¯
¯αi, βj, . . . , γt
¾
where
αi∈©
u0+pu1+· · ·+p`−2u`−2¯
¯u0, u1, . . . , u`−2∈Tmª ,
βj ∈©
v0+pv1+· · ·+p`−3v`−3¯¯v0, v1, . . . , v`−3∈Tmª , and so on and
γt∈© w0¯
¯w0∈Tmª whereTm is the Teichmuller set defined in Section 2.2.
We see that, by (2.15), we have
NC0 (j, pe) =pm(k1+k2+···+k`)NS0(j, pe), 1≤e≤m. (2.17)
Now, suppose that we introduce a map µ on GR(p`, m) that reduces every element in GR(p`, m) modulo p`−1. Suppose R = µ¡
GR(p`, m)¢
. Then µ(S) becomes a linear code overR and, furthermore we have
NS0(j, pe) =Nµ(S)H (j/(pm−1), pe) (2.18)
whereNµ(S)H (i, pe) denotes the number of codewords that are inµ(S) with Hamming weights congruent toimodulo pe and 1≤e≤m. The equation (2.18) is true, because
w0(a)≡(pm−1)wH(µ(a)) (modpe)
for alla∈S and 1≤e≤m. Now applying the methods used in the proof of Theorem 2.11, one can however easily show that
νp¡
Nµ(S)H (i, pe)¢
≥
¹k1+k2+· · ·+k`−1−pe−1 (p−1)pe−1
º
, (2.19)
for alli= 0,1, . . . , pe−1. Now, the theorem follows from combining (2.8), (2.17),(2.18) and (2.19).