Proceedings of the 44th Symposium on Ring Theory and ...

O<strong>the</strong>r pro<strong>of</strong>s that finite fields have <strong>the</strong> extension property with respect to <strong>the</strong> Hamming
weight have been given by Bogart, Goldberg, and Gordon [5] and by Ward and Wood
[29]. We will not prove <strong>the</strong> finite field case separately, because it is a special case <strong>of</strong> <strong>the</strong>
main <strong>the</strong>orem <strong>of</strong> this section:
Theorem 31. Let R be a finite ring. Then R has <strong>the</strong> extension property with respect to
<strong>the</strong> Hamming weight if and only if R is Frobenius.
One direction, that finite Frobenius rings have <strong>the</strong> extension property, first appeared in
[31, Theorem 6.3]. The pro<strong>of</strong> (which will be given in subsection 5.2) is based on <strong>the</strong> linear
independence <strong>of</strong> characters and is modeled on <strong>the</strong> pro<strong>of</strong> in [29] <strong>of</strong> <strong>the</strong> finite field case. A
combinatorial pro<strong>of</strong> appears in work <strong>of</strong> Greferath and Schmidt [12]. More generally yet,
Greferath, Nechaev, and Wisbauer have shown that <strong>the</strong> character module <strong>of</strong> any finite
ring has <strong>the</strong> extension property for <strong>the</strong> homogeneous and <strong>the</strong> Hamming weights [11]. Ideas
from this latter paper greatly influenced <strong>the</strong> work presented in subsection 5.4.
The o<strong>the</strong>r direction, that only finite Frobenius rings have <strong>the</strong> extension property, first
appeared in [32]. That paper carried out a strategy due to Dinh and López-Permouth [8].
Additional relevant material appeared in [33].
The rest <strong>of</strong> this section will be devoted to <strong>the</strong> pro<strong>of</strong> <strong>of</strong> Theorem 31.
5.2. Frobenius is Sufficient. In this subsection we prove half <strong>of</strong> Theorem 31, that a
finite Frobenius ring has <strong>the</strong> extension property, following <strong>the</strong> treatment in [31, Theorem
6.3].
Assume C 1 , C 2 ⊂ R n are two left linear codes, and assume f : C 1 → C 2 is an R-linear
isomorphism that preserves <strong>the</strong> Hamming weight. We want to show that f extends to
a monomial transformation <strong>of</strong> R n . The core idea is to express <strong>the</strong> weight-preservation
property <strong>of</strong> f as an equation <strong>of</strong> characters <strong>of</strong> C 1 and to use <strong>the</strong> linear independence <strong>of</strong>
characters to match up terms.
Let pr 1 , . . . , pr n : R n → R be <strong>the</strong> coordinate projections, so that pr i (x 1 , . . . , x n ) = x i ,
(x 1 , . . . , x n ) ∈ R n . Let λ 1 , . . . , λ n denote <strong>the</strong> restrictions <strong>of</strong> pr 1 , . . . , pr n to C 1 ⊂ R n .
Similarly, let µ 1 , . . . , µ n : C 1 → R be given by µ i = pr i ◦f. Then λ 1 , . . . , λ n , µ 1 , . . . , µ n ∈
Hom R (C 1 , R) are left R-linear functionals on C 1 . It will suffice to prove <strong>the</strong> existence
<strong>of</strong> a permutation σ <strong>of</strong> {1, . . . , n} and units u 1 , . . . , u n <strong>of</strong> R such that µ i = λ σ(i) u i , for
i = 1, . . . , n.
For any x ∈ C 1 , <strong>the</strong> Hamming weight <strong>of</strong> x is given by wt(x) = ∑ n
i=1 wt(λ i(x)), while
<strong>the</strong> Hamming weight <strong>of</strong> f(x) is given by wt(f(x)) = ∑ n
i=1 wt(µ i(x)). Because f preserves
<strong>the</strong> Hamming weight, we have
n∑
n∑
(5.1)
wt(λ i (x)) = wt(µ i (x)).
i=1
Using Lemma 24, observe that 1 − wt(r) = (1/|R|) ∑ π∈ ̂R
π(r), for any r ∈ R. Apply this
observation to (5.1) and simplify:
n∑ ∑
n∑ ∑
(5.2)
π(λ i (x)) = π(µ i (x)), x ∈ C 1 .
i=1 π∈ ̂R
i=1
i=1 π∈ ̂R
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