Weight enumeration of codes from finite spaces Relinde Jurrius - - PowerPoint PPT Presentation

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Weight enumeration of codes from finite spaces

Relinde Jurrius

Eindhoven University of Technology, The Netherlands

Finite Geometries, Third Irsee conference, June 19–25, 2011

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Outline

Projective systems and weight enumeration Generalized weight enumerator Codes form finite spaces Finite projective space: simplex code Finite affine space: 1-st order Reed-Muller code Extended weight enumerator Further questions and applications

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Projective systems

projective system n-tuple P = (P1, . . . , Pn) points Pj ∈ PG(r, q) in general position Matrix GP with coordinates of points of P as columns generates linear [n, r + 1] code.   

• equiv. classes of

linear [n, k] codes

• ver GF(q)

   ← →   

• equiv. classes of
• proj. systems of length n
• ver PG(k − 1, q)

  

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Weights in linear codes

✞ ✝ ☎ ✆

1 × k k × n 1 × n message m generator matrix G codeword c

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Weights in linear codes

✞ ✝ ☎ ✆

1 × k k × n 1 × n message m generator matrix G codeword c

Theorem

cj = 0 ⇐ ⇒ Pj is in nullspace of m We can determine weights by counting points Pj on hyperplanes.

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Generalized weight enumerator

For a subcode D ⊆ C we define support supp(D) union of the support of all words in D zero set zero(D) complement of support, i.e., all coordinates that are always zero weight wt(D) size of the support

Generalized weight enumerators

Polynomials counting for every dimension the number of subcodes

• f a given weight:

W (r)

C (X, Y ) =

• D∈C

dim(D)=r

X|zero(D)|Y |supp(D)|

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Weight enumeration of subcodes

✞ ✝ ☎ ✆ ✞ ✝ ☎ ✆

r × k k × n r × n nullspace = Π generator matrix G generates D Π ⊆ PG(k − 1, q) codim(Π) = r ← → D ⊆ C dim(D) = r

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Weight enumeration of subcodes

✞ ✝ ☎ ✆ ✞ ✝ ☎ ✆

r × k k × n r × n nullspace = Π generator matrix G generates D Π ⊆ PG(k − 1, q) codim(Π) = r ← → D ⊆ C dim(D) = r

Theorem

j ∈ zero(D) ⇐ ⇒ Pj ∈ Π

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Codes from finite spaces

Let P contain all points in PG(s − 1, q). The corresponding code is the simplex code Sq(s). It has length qs − 1 q − 1 and dimension s.

Example

S2(3) has generator matrix GP =   1 1 1 1 1 1 1 1 1 1 1 1  

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Codes from finite spaces

Π ⊆ PG(k − 1, q) codim(Π) = r ← → D ⊆ C dim(D) = r

• j ∈ zero(D) ⇐

⇒ Pj ∈ Π

• P contains all points in PG(s − 1, q)
• |zero(D)| = |Π| for all D

Theorem

W (r)

Sq(s)(X, Y ) =

s r

• q X(qs−r−1)/(q−1)Y (qs−qs−r)/(q−1)

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Codes from finite spaces

Let P contain all points in AG(s − 1, q), viewed as all points in PG(s − 1, q) not on a hyperplane H. The corresponding code is the 1-st order Reed Muller code RMq(1, s − 1). It has length qs−1 and dimension s.

Example

RM2(1, 3) has generator matrix GP =     1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1    

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Codes from finite spaces

Π ⊆ PG(k − 1, q) codim(Π) = r ← → D ⊆ C dim(D) = r

• j ∈ zero(D) ⇐

⇒ Pj ∈ Π

• if Π ⊆ H, no Pj is in Π, so wt(D) = n
• if Π ⊆ H, all of the Pj in Π form a subspace of AG(s − 1, q)
• f codimension r

Theorem

W (r)

RMq(1,s−1)(X, Y ) =

s − 1 r − 1

• q

Y n + qr s − 1 r

• q

Xqs−1−rY qs−1−qs−1−r

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Extended weight enumerator

For every linear [n, k] code C with generator matrix G we have: Extension code [n, k] code C ⊗ GF(qm) over some extension field GF(qm) generated by the words of C. Generator matrix All the extension codes of C have the same generator matrix G.

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Extended weight enumerator

For every linear [n, k] code C with generator matrix G we have: Extension code [n, k] code C ⊗ GF(qm) over some extension field GF(qm) generated by the words of C. Generator matrix All the extension codes of C have the same generator matrix G.

Extended weight enumerator

Polynomial counting “for all extension codes” the number of words

• f a given weight:

WC(X, Y, T) =

n

• w=0

Aw(T)Xn−wY w. So for T = qm we have WC(X, Y, qm) = WC⊗GF(qm)(X, Y ).

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Extended weight enumerator

The etended weight enumerator is completely determined by the set of generalized weight enumerators (and vice versa):

Theorem

WC(X, Y, T) =

k

• r=0

 

r−1

• j=0

(T − qj)   W (r)

C (X, Y ).

Moreover, their sets of supports are the same.

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Supports

Theorem (Simplex code)

Let c ∈ Sq(s) ⊗ GF(qm) with wt(c) = (qs − qs−r)/(q − 1), r < m. Then the points in P indexed by zero(c) are all the points in a subspace of PG(s − 1, q) of codimension r.

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Supports

Theorem (Simplex code)

Let c ∈ Sq(s) ⊗ GF(qm) with wt(c) = (qs − qs−r)/(q − 1), r < m. Then the points in P indexed by zero(c) are all the points in a subspace of PG(s − 1, q) of codimension r.

Theorem (1-st order Reed-Muller code)

Let c ∈ RMq(1, s) ⊗ GF(qm) with wt(c) = qs−1 − qs−1−r, r < m. Then the points in P indexed by zero(c) are all the points in a subspace of AG(s − 1, q) of codimension r.

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Supports

Theorem (Simplex code)

Let c ∈ Sq(s) ⊗ GF(qm) with wt(c) = (qs − qs−r)/(q − 1), r < m. Then the points in P indexed by zero(c) are all the points in a subspace of PG(s − 1, q) of codimension r.

Theorem (1-st order Reed-Muller code)

Let c ∈ RMq(1, s) ⊗ GF(qm) with wt(c) = qs−1 − qs−1−r, r < m. Then the points in P indexed by zero(c) are all the points in a subspace of AG(s − 1, q) of codimension r. So: the sets zero(c) for all codewords contains the incidence design

• f a finite geometry.

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Further questions and applications

• Weight enumeration of higher order Reed-Muller codes

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Further questions and applications

• Weight enumeration of higher order Reed-Muller codes
• Generalized Hamming weights of Reed-Muller codes

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Further questions and applications

• Weight enumeration of higher order Reed-Muller codes
• Generalized Hamming weights of Reed-Muller codes
• Link with perfect matroid designs and associated polynomials

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Further questions and applications

• Weight enumeration of higher order Reed-Muller codes
• Generalized Hamming weights of Reed-Muller codes
• Link with perfect matroid designs and associated polynomials
• Two weight codes: quadratic extension code of simplex code

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Further questions and applications

• Weight enumeration of higher order Reed-Muller codes
• Generalized Hamming weights of Reed-Muller codes
• Link with perfect matroid designs and associated polynomials
• Two weight codes: quadratic extension code of simplex code
• Dimension of a design (generalization of p-rank)

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Thank you for your attention.