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Upper Bounds

forRing-Linear

Codes

Eimear Byrne,Marcus

Greferath,Axel Kohnert,

Vitaly

Skachek

Upper Bounds for Ring-Linear Codes

Eimear Byrne1, Marcus Greferath1, Axel Kohnert2,

Vitaly Skachek1

1Claude Shannon Institute andSchool of Mathematical Sciences

University College DublinIreland

2Dept MathematicsUniversity of Bayreuth

Germany

May 19 2009

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Upper Bounds

forRing-Linear

Codes

Eimear Byrne,Marcus


Vitaly

Skachek

Outline

Codes over finite fields

Code optimalityBounds for codes for the Hamming weight

Ring-linear coding

The homogeneous weight

Bounds on the size of a code for the homogeneous weight

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Upper Bounds

forRing-Linear

Codes

Eimear Byrne,Marcus


Vitaly

Skachek

Notation

F = GF (q ), q = p m some prime p

R is a finite ring with identity

R̂ := HomZ(R ,C×) the characters on (R , +)

χ ∈ R̂ is a character on (R , +)

C is a code of length n and minimum distance d

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Upper Bounds

forRing-Linear

Codes

Eimear Byrne,Marcus


Vitaly

Skachek

Using Codes for Error Correction

One parameter that indicates the error-correcting capability of a code is its minimum distance.

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Upper Bounds

forRing-Linear

Codes

Eimear Byrne,Marcus


VitalySkachek


One parameter that indicates the error-correcting capability of a code is its minimum distance.The higher the minimum distance, the more errors that can bedetected and corrected by the receiver.

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Upper Bounds

forRing-Linear

Codes

Eimear Byrne,Marcus


VitalySkachek



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Upper Bounds

forRing-Linear

Codes

Eimear Byrne,Marcus


VitalySkachek

Code Optimality

The Main Coding Problem:

1 For fixed length n and minimum distance d , what is themaximum size of any code over R ?i.e., what is AR (n, d )?

2 For a fixed length n and minimum distance d , what is the

maximum size of any linear code over R ?i.e., what is B R (n, d )?

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Upper Bounds

forRing-Linear

Codes

Eimear Byrne,Marcus


VitalySkachek

Some Distance Functions

Definition (Hamming Metric)

Let u, v ∈ R n. The Hamming distance between u and v is thenumber of components where u and v differ, i.e.

dHam(u, v) = |{i : u i = v i }|

u = [0, 0, 1, 1, 3, 3], v = [1, 2, 2, 1, 1, 3] ∈ Z4

d Ham(u, v) = 4.

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Upper Bounds

forRing-LinearCodes

Eimear Byrne,Marcus


VitalySkachek

Some Distance Functions

Definition (Lee Metric)

Let u , v ∈ Zm. The Lee distance between u and v is theabsolute value modulo m of u − v , i.e.

dLee(u , v ) = |u − v |m =

u − v if u − v ∈ {0, ..., m/2}v − u otherwise

If u, v ∈ Znm then dLee(u, v) =

i =1..n |u i − v i |m.

u = [0, 0, 1, 1, 3, 3], v = [1, 2, 2, 1, 1, 3] ∈ Z4

d Lee(u, v) = 1 + 2 + 1 + 2 = 6

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Upper Bounds

forRing-LinearCodes

Eimear Byrne,Marcus


VitalySkachek

Some Bounds for Codes over Finite Fields

Singleton: |C | ≤ Aq (n, d ) ≤ q n−d +1

Hamming: |C | ≤ Aq (n, d ) ≤ q n

V q (n, d −12

),

Plotkin: |C | ≤ Aq (n, d ) ≤ d d −γ n

, γ = q −1q

, if n < d γ

Gilbert-Varshamov: Aq (n, d ) ≥ q n

V q (n,d −1)

Elias-Bassalygo bound

Mc-Eliece-Rodemich-Rumsey-Welch boundLinear Programming bound

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Upper Bounds

forRing-LinearCodes

Eimear Byrne,Marcus


VitalySkachek

Asymptotic Representations

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Upper Bounds

forRing-LinearCodes

Eimear Byrne,Marcus


VitalySkachek

Codes over Finite Rings

Definition

An code of length n over R is a nonempty subset of R n. A(left) linear code of length n over R is a left R -submodule of R n.

We will usually assume that R is a finite Frobenius ring.

Many of the foundational results of classical coding theory (e.g.the MacWilliams’ theorems) can be extended to the finite ringcase when R is Frobenius.

[Wood, Honold, Nechaev, Greferath, Schmidt..]

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Upper Bounds

forRing-LinearCodes

Eimear Byrne,Marcus


VitalySkachek

Finite Frobenius Rings

For a finite ring R , R̂ is an R − R bimodule via

χr (x ) = χ(rx ), r χ(x ) = χ(xr )

for all x , r ∈ R , χ ∈ R̂ .R is a finite Frobenius ring iff

R R R R̂

Then R R̂ = R χ for some (left) generating character χ

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Upper Bounds

forRing-LinearCodes

Eimear Byrne,Marcus


VitalySkachek

Finite Frobenius Rings

Let R and S be finite Frobenius rings, let G be a finite group.The following are examples of Frobenius rings.

integer residue rings Zm

Galois rings

principal ideal rings

R × S

the matrix ring M n(R )the group ring R [G ]

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Upper Bounds

forRing-LinearCodes

Eimear Byrne,Marcus


VitalySkachek

Homogeneous Weights

Definition

A weight w : R −→ Q is (left) homogeneous , if w (0) = 0 and

1 If Rx = Ry then w (x ) = w (y ) for all x , y ∈ R .

2 There exists a real number γ such that

y ∈Rx

w (y ) = γ |Rx | for all x ∈ R \ {0}.

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Upper Bounds

forRing-LinearCodes

Eimear Byrne,Marcus


VitalySkachek

Examples of Homogeneous Weights

ExampleOn every finite field Fq the Hamming weight is a homogeneousweight of average value γ = q −1

q .

Example

On Z4 the Lee weight is homogeneous with γ = 1.

x 0 1 2 3

w Lee(x ) 0 1 2 1

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Upper Bounds

forRing-LinearCodes

Eimear Byrne,Marcus


VitalySkachek


ExampleOn Z10 the following weight is homogeneous with γ = 1:

x 0 1 2 3 4 5 6 7 8 9

whom(x ) 0 1 54 1 5

4 2 54 1 5

4 1

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Upper Bounds

forRing-LinearCodes

Eimear Byrne,Marcus


VitalySkachek


Example

On the ring R of 2 × 2 matrices over GF(2) the weight

w : R −→ R, X →

0 : X = 0,2 : X singular, X = 0,1 : otherwise,

is a homogeneous weight of average value γ = 32 .

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Upper Bounds

forRing-LinearCodes

Eimear Byrne,Marcus


VitalySkachek


Example

On a local Frobenius ring R with q -element residue field theweight

w : R −→ R, x →

0 : x = 0,q

q −1 : x ∈ soc (R ), x = 0,

1 : otherwise,

is a homogeneous weight of average value γ = 1.

Which finite rings admit a homogeneous weight?

Up to the choice of γ , every finite ring admits a uniquehomogeneous weight .

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Upper Bounds

forRing-LinearCodes

Eimear Byrne,Marcus


VitalySkachek

Homogeneous Weights of FFRs

Theorem (Honold)

Let R be a finite Frobenius ring with generating character χ.

Then the homogeneous weights on R are precisely the functions

w : R −→ R, x → γ

1 −1

|R ×|

u ∈R ×

χ(xu )

where γ is a real number.

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Upper Bounds

forRing-LinearCodes

Eimear Byrne,Marcus


VitalySkachek

Bounds on AR (n, d ) for the Homogeneous Weight

The following bounds have been found for codes over FFRs forthe homogeneous weight.

Sphere-packing (Hamming)

Sphere-covering (Gilbert-Varshamov)

Plotkin-like bounds

Elias-like bounds

Singleton-like boundLinear programming bound

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Upper Boundsfor

Ring-LinearCodes

Eimear Byrne,Marcus


VitalySkachek

A Key Lemma

Lemma

Let C ≤ R R n

be a linear code, and let x ∈ R n

. Then1

|C |

c ∈C

w (x + c ) = γ |supp(C )| +

i ∈supp(C )

w (x i ).

C

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Upper Boundsfor

Ring-LinearCodes

Eimear Byrne,Marcus


VitalySkachek

Residual Codes

Definition

Let C ≤R R n, c ∈ R n. Res (C , c ) := {(x i ) : x ∈ C , c i = 0}.Example

Let C be the Z4-linear code generated by

1 0 0 0 3 1 2 10 1 0 0 1 2 3 10 0 1 0 3 3 3 20 0 0 1 2 3 1 1

.

Let c = [0, 0, 0, 2, 0, 2, 2, 2]. Then Res (C , c ) is generated by

1 0 0 30 1 0 10 0 1 3

0 0 0 2

.

R id l C d

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Upper Boundsfor

Ring-LinearCodes

Eimear Byrne,Marcus


VitalySkachek

Residual Codes

Theorem

Let C ≤ R R n have minimum homogeneous weight d, and let c ∈ C satisfy (c ) := w Ham < d

γ . Then Res (C , c ) has

length n − (c ),

minimum homogeneous weight d ≥ d − γ(c ),

|Res (C , c )| =|C |

|Rc |and

|C | ≤ |Rc | d − γ(c )d − γ n

.

B d B ( d) f h H W i h

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Upper Boundsfor

Ring-LinearCodes

Eimear Byrne,Marcus


VitalySkachek

Bounds on B R (n, d ) for the Homogeneous Weight

Corollary (BGKS)

Let C ≤ R R

n

be a linear code of minimum homogeneous weight d and minimum Hamming weight where ≤ n ≤ d γ

. Then

|C | ≤ |R |d − γ

d − γ n.

B d B ( d) f h H W i h

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Upper Boundsfor

Ring-LinearCodes

Eimear Byrne,Marcus


VitalySkachek


Corollary (BGKS)

Let C ≤ R R n be a linear code of minimum homogeneous

weight d and minimum Hamming weight where < n ≤ d γ

.Let Q be the maximum size of any minimal ideal of R. Then

|C | ≤ Q d − γ

d − γ n.

A Pl tki O ti l C d

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Upper Boundsfor

Ring-LinearCodes

Eimear Byrne,Marcus


VitalySkachek

A Plotkin Optimal Code

ExampleLet R = F

2×22 . Let C be the length 16m − 1 Simplex Code over

R . Then |C | = 16m,

d = |R |mγ = 16mγ,

:= dHam(C ) = 16m −16m

4=

3

416m.

R has 3 minimal ideals, each of size Q = 4 and so

|C | ≤ Q d − γd − γ n

= 416mγ − 3

4 16mγ

16mγ − (16m − 1)γ = 4

16m

4= 16m.

B d B ( d) f th H W i ht

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Upper Boundsfor

Ring-LinearCodes

Eimear Byrne,Marcus


VitalySkachek


Singleton-like bounds:

Theorem (BGKS)

Let C ≤ R R n be an [n, d ] linear code and suppose that n ≤ d γ

.Then

n − |R | − 1

|R |

d

γ

≥

log|R | |C | − 1

.

Theorem (BGKS)

Let C be an [n, d ] code over R satisfying n ≤d γ and (C ) < n.

Let P := max{|Ra| : a ∈ R n, Ra ≤ C , (a) < n}. Then

n −

P − 1

P

d

γ

≥ logP |C | − logP |R | .

A Class of MDS Codes

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Upper Boundsfor

Ring-LinearCodes

Eimear Byrne,Marcus


VitalySkachek

A Class of MDS Codes

Example

Let R be a chain ring of length 2. Then R × = R \rad R and|R | = q 2. Let U := R 2\rad R 2, let P := {xR : x ∈ U }. Then

|P| = q 2

+ q .Let C < R R n be the length n := q 2 + q code with 2 × ngenerator matrix whose columns are the distinct elements of P .

Clearly (c ) < n for each c ∈ C .

C is free of rank 2 and the maximal cyclic submodules of C have size P := |R | = q 2.

Let r = logP |C | − 1 = logq 2 q 4 − 1 = 1.

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Upper Boundsfor

Ring-LinearCodes

Eimear Byrne,Marcus


VitalySkachek

Example (cont.)

Setting γ = 1, each word xG of C has weight

w (xG ) =

q 2 + q if x ∈ U q 3

q −1 if x ∈ radR 2,

=⇒ n −

P − 1

P d

= n −

q 2 − 1

q 2(q 2 + q )

= n − q 2 + q − 1 −1

q

= q 2 + q − q 2 − q + 1 = 1 = r .

References

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Upper Boundsfor

Ring-LinearCodes

Eimear Byrne,Marcus


VitalySkachek

References

E. Byrne, M. Greferath and M. E. O’Sullivan, The linear programming bound for codes over finite Frobenius rings,Designs, Codes and Cryptography, Vol. 42 , 3 (2007), pp.289 - 301.

I. Constantinescu and W. Heise, A metric for codes over

residue class rings of integers , Problemy PeredachiInformatsii 33 (1997), no. 3, 22–28.

M. Greferath and M. E. O’Sullivan, On Bounds for Codes over Frobenius Rings under Homogeneous Weights ,Discrete Mathematics 289 (2004), 11–24.

T. Honold, A characterization of finite Frobenius rings,Arch. Math. (Basel), 76 (2001), 406–415.

J. A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121

(1999), 555–575.

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