Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Thomas
Honold
Geometric
View of Linear
Codes
Linear Codes over Finite Chain
Rings—Algebraic Theory
Finite Chain
Rings
Modules over
Finite Chain
Rings
Thomas Honold
Institute of Information and Communication Engineering
Zhejiang University
CIMPA Summer School
August 2008
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Outline
Thomas
Honold
Geometric
View of Linear
Codes
Finite Chain
Rings
Modules over
Finite Chain
Rings
1 Geometric View of Linear Codes
2 Finite Chain Rings
3 Modules over Finite Chain Rings
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Thomas
Honold
Geometric
View of Linear
Codes
Finite Chain
Rings
Modules over
Finite Chain
Rings
Today’s Lecture: #2
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Thomas
Honold
Geometric
View of Linear
Codes
Proposition
(i) For 1 ≤ i ≤ m we have dimR/N θi−1 M/θi M =
dimR/N M[θi ]/M[θi−1 ] = dimR/N M[θ] ∩ θi−1 M .
(ii) The integers µi := dimR/N M[θ] ∩ θi−1 M , 1 ≤ i ≤ m
(Ulm-Kaplansky invariants of R M) satisfy µ1 ≥ µ2 ≥ · · · ≥ µm
and µ1 + µ2 + · · · + µm = logq |M|.
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Proof.
(i) The map θi−1 M → θi M, x 7→ θx induces
an (additive)
isomorphism θi−1 M/ M[θ] ∩ θi−1 M ∼
= θi M. Hence
i−1 θ M = θi M and θi−1 M/θi M = M[θ] ∩ θi−1 M .
|M[θ] ∩ θi−1 M|
Similarly, M[θi ] → M[θ], x 7→ θi−1 x induces an isomorphism
M[θi ]/M[θi−1 ] → M[θ] ∩ θi−1 M, proving the second equality.
(ii) The inequality µi ≥ µi+1 follows from µi = dimR/N (Vi ) and
Vi ⊇ VQ
i+1 . Part
i−1(i) andi the
upper Loewy series give
θ M/θ M = q µ1 +···+µm .
|M| = m
i=1
Linear Codes
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Example
Let R = Z4 = {0, 1, 2, 3}, M = Zn4 , θ = 2 (the only choice for θ).
The possible periods (heights) of x ∈ Zn4 are 1, 2, 4 (resp. 0, 1, 2).
x ∈ Zn4 has period 1 (height 2) iff x = 0
x has period 2 (height 1) iff 2x = 0 ∧ x 6= 0 iff xi ∈ {0, 2} and at
least one xi = 2.
x has period 4 (height 0) iff at least one xi ∈ {1, 3}.
We have Zn4 [2] = 2Zn4 , so both Loewy series coincide (a property
of free modules in general).
The factors are
2Zn4 ∼
= Zn2 (divide the entries of x by 2)
n
Z4 /2Zn4 ∼
= Zn2 (read the entries of x modulo 2)
Linear Codes
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Geometric
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Example
Let C be the linear code of even length n over Z4 generated by


1 1
1 ... 1
2 2
0 . . . 0



.. 
..
0 2

.
2
.


. .

.
.
..
..
. . 0
 ..
0 ... 0
2 2
2C = {00 . . . 0, 22 . . . 2}
C[2] is either 2Zn4 (if n is odd) or the even-weight subcode of 2Zn4
(if n is even)
So the Loewy series C ⊃ 2C ⊃ {0} and C ⊃ C[2] ⊃ {0} are
different, but its factors C/2C ∼
= C[2] and C/C[2] ∼
= 2C are the
same.
Linear Codes
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Thomas
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Definition
Partitions of integers
An (integer) partition is a sequence λ = (λ1 , λ2 , . . . ) with λi ∈ Z,
λ1 ≥ λ2 ≥ · · · , and λi = 0 for all but finitely many i. The numbers
λi > 0 are called the parts of λ, and |λ| = λ1 + λ2 + · · · is called
the weight of λ. If |λ| = n, we say that λ is an (unordered)
partition of n and write λ ⊢ n.
Trailing zeros are usually suppressed, e.g. (2, 1, 1, 0, 0, . . . ) is
written as (2, 1, 1), or as 4 = 2 + 1 + 1.
A partition λ is often visualized by an (empty) Young tableaux Tλ ,
shown here for λ = (6, 6, 4, 3, 2, 2, 1, 1, 1) ⊢ 26:
Think of Tλ as the union of all unit squares in the Euclidean plane
whose upper right corners have coordinates (i, j) where i ≥ 1 and
1 ≤ j ≤ λi .
Linear Codes
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Chain Rings—
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Thomas
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Definition
Let λ be a partition. The conjugate λ′ of λ is the partition whose
Young tableaux Tλ′ is obtained from Tλ by a reflection at the line
y = x.
Properties
• If λ ⊢ n then also λ′ ⊢ n.
• The parts of λ′ are λ′i = |{j; λj ≥ i}|.
• The largest part λ′1 of λ′ is equal to the number of parts of λ.
Example
The conjugate of λ = (6, 6, 4, 3, 2, 2, 1, 1, 1) is λ′ = (9, 6, 4, 3, 2, 2).
Linear Codes
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The Module Classification Theorem
Thomas
Honold
Geometric
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Theorem
For any finite module R M there exists a uniquely determined
partition λ ⊢ logq |M| into parts λi ≤ m such that
RM
∼
= R/N λ1 ⊕ R/N λ2 ⊕ · · · ⊕ R/N λr .
(Here r denotes the number of parts of λ.)
Remarks
• The theorem holds mutatis mutandis for right modules MR .
• The theorem says in particular that every finite R-module is a
direct sum of cyclic R-modules.
Linear Codes
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Proof of the theorem.
Lr
Uniqueness part: Suppose R M ∼
= i=1 R/N λi .
The upper Loewy series of R/N λi is clearly
R/N λi ) N/N λi ) N 2 /N λi ) · · · ) N λi /N λi = {0}
with all successive quotients 1-dimensional over R/N.
Hence
dimR/N θj−1 M/θj M = |{i; λi ≥ j}| = λ′j
This shows that λ′ (and hence λ) is uniquely determined by R M.
Linear Codes
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Existence part: Since any cyclic R-module is isomorphic to R/N i
for some i ≤ m, it suffices to prove that R M is a direct sum of
cyclic R-modules. We use induction on |M|.
Assume M 6= {0} (otherwise the assertion is clear) and choose
x1 ∈ M of maximal period θλ1 . By induction,
M/Rx1 = Ry2 ⊕ · · · ⊕ Ryr .
(⋆)
Let xi ∈ M such that yi = xi + Rx1 (2 ≤ i ≤ r ). Then
M = Rx1 + Rx2 + · · · + Rxr .
(⋆⋆)
To make (⋆⋆) a direct sum decomposition, the xi ’s must be chosen
in a special way.
Suppose yi has period θλi , i.e. θλi xi ∈ Rx1 and λi is the smallest
such integer.
There exists r ∈ R such that θλi xi = θλi ax1 . (Otherwise the period
of x1 would be smaller than the period of xi .)
Then θλi (xi − ax1 ) = 0, and so xi − ax1 has period θλi .
Replacing xi by xi − ax1 , we may assume that the xi ’s in (⋆⋆) have
the same period θλi as the yi ’s in (⋆).
Linear Codes
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Proof cont’d.
Now suppose
a1 x1 + a2 x2 + · · · + ar xr = 0 where ai ∈ R
Modulo Rx1 this reads a2 y2 + · · · + ar yr = 0, and (⋆) gives
a2 y2 = · · · = ar yr = 0.
Since xi has the same period as yi , we conclude
a2 x2 = · · · = ar xr = 0, and further a1 x1 = 0.
Linear Codes
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Thomas
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Definition
Lr
The partition λ ⊢ logq |M| such that R M ∼
= i=1 R/N λi is called
the shape (or type) of R M. The conjugate
partition λ′ , which
satisfies λ′i = dimR/N θi−1 M/θi M , is called the conjugate shape
of R M. The integer r = λ′1 (number of nonzero summands in a
direct sum decomposition of R M into cyclic R-modules) is called
the rank of R M (and denoted by rk M).
Definition
A sequence x1 , x2 , . . . , xs of elements of R M is said to
(i) independent if a1 x1 + a2 x2 + · · · + as xs = 0 with ai ∈ R
implies ai xi = 0 for 1 ≤ i ≤ s;
(ii) a basis of R M if x1 , . . . , xr are independent and generate R M;
(iii) linearly independent if a1 x1 + a2 x2 + · · · + as xs = 0 with
ai ∈ R implies ai = 0 for 1 ≤ i ≤ s.
Remarks
• Properties (ii) is equivalent to R M = Rx1 ⊕ · · · ⊕ Rxs .
• Property (iii) is equivalent to (i) and xi ∈ M × for 1 ≤ i ≤ s.
Linear Codes
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Chain Rings—
Algebraic
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Thomas
Honold
Geometric
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Example
Consider the linear codes C3 , C4 over Z4 generated by




1 1 1 1
1 1 1
2 2 0 0

G3 = 2 2 0 resp. G4 = 
0 2 2 0 .
0 2 2
0 0 2 2
The rows of G3 are independent, hence a basis of C3 .
The first 3 rows of G4 form a basis of C4 (since they are
independent, and the last row is a linear combination of the 1st
and 2nd row).
Hence both modules have the same shape λ = (2, 1, 1) with
Young diagram
rank equal to 3 and cardinality |C3 | = |C4 | = 22+1+1 = 16.
Linear Codes
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Thomas
Honold
Geometric
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Example
The linear code over Z4 generated by


1 0 0 1 2 3 1
0 1 0 1 1 2 3
0 0 1 1 3 1 2
has shape λ = (2, 2, 2) with Young diagram
rank 3 and cardinality 22+2+2 = 64.
Linear Codes
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Thomas
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Geometric
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Free Modules
Recall that R M is free if R M ∼
= R k for some integer k ≥ 0. (The
integer k is the rank of M.) Equivalently, R M has “rectangular”
shape (m, m, . . . , m).
Free modules are important, since they are the ambient spaces
for both linear codes and projective Hjelmslev geometries over R
(→ Ivan Landjev’s lecture).
Definition
Let R M be free of rank k . The projective geometry PG(R M) is the
lattice of all submodules of R M. A free submodule U ≤ R M of
rank 1 (rank k − 1, rank s + 1) is said to be a point (resp.
hyperplane, s-flat) of PG(R M).
Important problems
• Determine the number of points (hyperplanes, s-flats, or in
general λ-shaped submodules) of PG(R M).
• How many hyperplanes contain a given point (s-flat, or
λ-shaped submodule)?
Linear Codes
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Thomas
Honold
Theorem
Suppose R M is free of rank k , and U is a (not necessarily free)
submodule of R M of shape λ and rank s.
1
For every basis x1 , . . . , xs of U there exists a basis y1 , . . . , yk
of M such that xi ∈ Ryi for 1 ≤ i ≤ s. (Bases of U and M
related in this way are called stacked bases.)
2
The quotient module M/U has shape
(m − λk , m − λk −1 , . . . , m − λ1 ).
3
If U ∗ 6= ∅ then U is the sum of its free rank 1 submodules
(i.e., points).
4
If (M/U)∗ 6= ∅ then U is the intersection of all free rank k − 1
submodules of R M (i.e., hyperplanes) containing U.
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Remark
Part (2) says that the shape of M/U is
the complement of λ in the
k × m-rectangle corresponding to M.
M/U
U
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Thomas
Honold
Geometric
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Lemma
If R M is free then M[θi ] = θm−i M for 0 ≤ i ≤ m.
Proof.
Suppose M is free of rank k . For the shape λ of R M this means
λ1 = · · · = λk = m, λ′1 = · · · = λ′m = k .
′
i
]/M[θi−1 ] = q λi = q k for 1 ≤ i ≤ m, and hence
So M[θ
M[θi ] = q ki for 0 ≤ i ≤ m.
By a similar
argument using the upper Loewy series of R M we
have θi M = q k (m−i) .
Hence M[θi ] = θm−i M for 0 ≤ i ≤ m. Together with
θm−i M ⊆ M[θi ] this gives M[θi ] = θm−i M for 0 ≤ i ≤ m.
Remark
Suppose R is a proper chain ring (i.e. m ≥ 2). If there exists
i ∈ {1, 2, . . . , m − 1} such that M[θi ] = θm−i M, then R M is
necessarily free.
Linear Codes
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Proof of the theorem.
(1) Suppose M is free of rank k , U ≤ M has shape λ, and
x1 , . . . , xs is a basis of U with xi of period θλi .
M[θλi ] = θm−λi M ⇒ ∃yi ∈ M with xi = θm−λi yi (1 ≤ i ≤ s).
y1 , . . . , ys are linearly independent (since their period is θm ),
hence may be extended to a basis y1 , . . . , yk of M (e.g., by first
extending θm−1 y1 , . . . , θm−1 ys to a basis of the R/N-space M[θ]
and then proceeding as before).
(2) With yi as in (1) we have
M/U ∼
= Ry1 /Rx1 ⊕ · · · ⊕ Rys /Rxs ⊕ R k −s
∼
= R/N m−λ1 ⊕ · · · ⊕ R/N m−λs ⊕ R/N m ⊕ · · · ⊕ R/N m
= R/N m−λ1 ⊕ · · · ⊕ R/N m−λk .
(3) U × 6= ∅ is equivalent to λ1 = m, i.e. x1 ∈ M × .
For j ≥ 2, if xj ∈
/ U × then x1 + xj ∈ U × , so U is generated by U × .
(4) (M/U)∗ 6= ∅ is equivalent to λk = 0. In this case U is
contained in the hyperplane H = Ry1 + · · · + Ryk −1 .
For z ∈ H \ U, z = r1 y1 + · · · + rk −1 yk −1 with θm−λj ∤ rj , say, let H ′
be the hyperplane obtained from H by replacing yj by yj + θλj yk .
One easily checks that U ⊆ H ′ , but z ∈
/ H ′.
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Thomas
Honold
Geometric
View of Linear
Codes
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G. J. Janusz.
Separable algebras over commutative rings.
Transactions of the American Mathematical Society,
122:461–479, 1966.
R. Raghavendran.
Finite associative rings.
Compositio Mathematica, 21:195–229, 1969.
J. A. Wood.
Duality for modules over finite rings and applications to
coding theory.
American Journal of Mathematics, 121(3):555–575,
1999.