Graph Codes Tom Hholdt, Jrn Justesen and Fernando Piero The - - PowerPoint PPT Presentation

Outline The Codes Bounds on the minimum distance Dimension of Graph Codes Special results when q is a square Performance

Graph Codes

Tom Høholdt, Jørn Justesen and Fernando Piñero The Technical University of Denmark Mumbay December 2013

Tom Høholdt, Jørn Justesen and Fernando Piñero The Technical University of Denmark Graph Codes

Outline The Codes Bounds on the minimum distance Dimension of Graph Codes Special results when q is a square Performance Product Codes

Code Construction

The bipartite graph G is used to define a code over Fq by associating a symbol with each edge and letting all edges that meet in a left vertex satisfy the parity checks of an (n, k1, d1) code C1 and edges that meet in a right vertex satisfy the parity checks of an (n, k2, d2) code C2. The resulting code is denoted by (G, C1 : C2). Thus the length of the code is N = mn

Dimension

K ≥ N − m(n − k1) − m(n − k2) so R ≥ r1 + r2 − 1

Adjacency Matrix

The bounds on the minimum distance involve the second largest eigenvalue of the adjacency matrix of the graph. Let x1, x2, . . . , xm be the left vertices V1 and y1, y2, . . . , ym the right vertices V2. Define the matrix M = mij where mij = 1 if xi is connected to yj else The adjacency matrix of the bipartite graph is then A =

• M

MT

Previous Results

Sipser and Spielman 1996 If d1 = d2 = d D ≥ dm d−λ

n−λ

Janwa and Lal 2003 If d1 ≥ d2 ≥ λ

2

D ≥ m

n (d1d2 − λ 2(d1 + d2))

Roth and Skachek 2006 D ≥ m

d1d2−λ√ d1d2 n−λ

New Results I

D ≥ md1 d2 − λβ n − λβ where β = λ(d1 − d2) +

• λ2(d1 − d2)2 + 4d1d2(n − d1)(n − d2)

2d1(n − d2)

The fundamental observation is Lemma Let v ∈ R2m where v⊥1 then < v, Av >≤ λ < v, v > where λ is the second largest eigenvalue of A.

Proof

Let ` E be a set of edges in G corresponding to the nonzero positions of a nonzero codeword of C. Let S be the subset of vertices in V1 incident with ` E and let T be the subset of vertices in V2 incident with `

• E. We will get

the bound on D from a bound of |` E|. Suppose that |S| = a and |T| = αa, α ≥ 1, and let e be the average valency of the vertices in S, thus e

α the average valency in T. Let

v = (vi) be a vector of length 2m where vi =        1 if i ∈ S −

a m−a

if i ∈ V1 \ S β if i ∈ T − αβa

m−αa

if i ∈ V2 \ T where 0 < β ≤ 1.

Proof

Let ` E be a set of edges in G corresponding to the nonzero positions of a nonzero codeword of C. Let S be the subset of vertices in V1 incident with ` E and let T be the subset of vertices in V2 incident with `

• E. We will get

the bound on D from a bound of |` E|. Suppose that |S| = a and |T| = αa, α ≥ 1, and let e be the average valency of the vertices in S, thus e

α the average valency in T. Let

v = (vi) be a vector of length 2m where vi =        1 if i ∈ S −

a m−a

if i ∈ V1 \ S β if i ∈ T − αβa

m−αa

if i ∈ V2 \ T where 0 < β ≤ 1.

Proof

Let ` E be a set of edges in G corresponding to the nonzero positions of a nonzero codeword of C. Let S be the subset of vertices in V1 incident with ` E and let T be the subset of vertices in V2 incident with `

• E. We will get

the bound on D from a bound of |` E|. Suppose that |S| = a and |T| = αa, α ≥ 1, and let e be the average valency of the vertices in S, thus e

α the average valency in T. Let

v = (vi) be a vector of length 2m where vi =        1 if i ∈ S −

a m−a

if i ∈ V1 \ S β if i ∈ T − αβa

m−αa

if i ∈ V2 \ T where 0 < β ≤ 1.

Proof

We can directly calculate v TAv =

2maβ (m−a)(m−αa)(me − naα)

The fundamental inequality and this result give

2maβ (m−a)(m−αa)(me − naα) ≤

λ(a + (m − a)

a2 (m−a)2 + aαβ2 + (m − αa) α2β2a2 (m−αa)2)

and this by a straightforward calculation leads to the following bound on a a ≥ m α 2eβ − λ(1 + αβ2) 2βn − λ(1 + β2) (1) which holds for any positive β.

Proof

The lower bound on a is met if and only if v is an eigenvector associated with the eigenvalue λ, i.e. Av = λv, and a necessary condition for this, where we

• nly look at the upper part of A is

λ = eβ − (n − e) aαβ m − aα and βλ = e α − (n − e α) a m − a These two conditions lead to the following expressions for a a = m eβ − λ α(βn − λ) (2) a = m

e α − βλ

n − βλ (3) and by eliminating a we get the equation for β. It can be seen that there is a positive solution less than 1.

Proof

The lower bound on a is met if and only if v is an eigenvector associated with the eigenvalue λ, i.e. Av = λv, and a necessary condition for this, where we

• nly look at the upper part of A is

λ = eβ − (n − e) aαβ m − aα and βλ = e α − (n − e α) a m − a These two conditions lead to the following expressions for a a = m eβ − λ α(βn − λ) (2) a = m

e α − βλ

n − βλ (3) and by eliminating a we get the equation for β. It can be seen that there is a positive solution less than 1.

Proof

Maximizing the right side of (1) with respect to β actually leads to the same equation. Thus this is the sharpest lower bound that can be obtained by this method. The bound can be met if there is a subgraph on S and T with exactly valencies e and e

α (which we expect will rarely be

the case). Since D ≥ ea the lower bound increases with a and e, we thus get a new lower bound by choosing α = e

d2

since the bound on a decreases with α and then choosing e = d1.

Proof

Maximizing the right side of (1) with respect to β actually leads to the same equation. Thus this is the sharpest lower bound that can be obtained by this method. The bound can be met if there is a subgraph on S and T with exactly valencies e and e

α (which we expect will rarely be

the case). Since D ≥ ea the lower bound increases with a and e, we thus get a new lower bound by choosing α = e

d2

since the bound on a decreases with α and then choosing e = d1.

Comparisons

If n = 16, λ = 4, d1 = 8, d2 = 4 the new bound gives D ≥ 4m

5

Janwa-Lal gives D ≥ m

2

and Roth-Skachek D ≥ 0.78m

New Results II

By a more careful analysis we get If d1 = d2 = d D ≥ da where

m a ≤ n + 1 − d + (n+1−d)(λ2−d2+2d−1) n+1+d2−d−λ2

New Results II

The graph of a generalized quadrangle over F8 has m = 585, n = 9, λ = 4. With C1 = C2 = RSext(9, 6, 4) The bound gives D ≥ 4 × 45 whereas the previous bounds give nothing.

A Question

Since all the bounds are decreasing functions of λ it is natural to ask: How small can λ be?

An Answer

A special case of a recent result ( Janwa +TH) Let G be a n-regular connected bipartite graph with m + m vertices then λ ≥

• mn−n2

r−1

where r is the rank of the transfer-matrix M and equality holds iff the eigenvalues of A are ±n with multiplicity 1,±λ with multiplicity r − 1 and 0 with multiplicity 2m − 2r.

An Answer

A special case of a recent result ( Janwa +TH) Let G be a n-regular connected bipartite graph with m + m vertices then λ ≥

• mn−n2

r−1

where r is the rank of the transfer-matrix M and equality holds iff the eigenvalues of A are ±n with multiplicity 1,±λ with multiplicity r − 1 and 0 with multiplicity 2m − 2r.

Duals of Graph Codes

Theorem (G, C1 : C2) = (G, C1 : Fn

q) ∩ (G, Fn q : C2)

(G, C1 : C2)⊥ = (G, C⊥

1 : Fn q) + (G, Fn q : C⊥ 2 ).

(G, C1 : Fn

q) is the Graph Code on G where the

codeword depends only on C1 and V1. (G, Fn

q : C2) is the Graph Code on G where the

codeword depends only on C2 and V2. Their intersection is (G, C1 : C2). (G, C1 : Fn

q)⊥ = (G, C⊥ 1 : Fn q) implies the second

statement.

Duals of Graph Codes

Theorem (G, C1 : C2) = (G, C1 : Fn

q) ∩ (G, Fn q : C2)

(G, C1 : C2)⊥ = (G, C⊥

1 : Fn q) + (G, Fn q : C⊥ 2 ).

(G, C1 : Fn

q) is the Graph Code on G where the

codeword depends only on C1 and V1. (G, Fn

q : C2) is the Graph Code on G where the

codeword depends only on C2 and V2. Their intersection is (G, C1 : C2). (G, C1 : Fn

q)⊥ = (G, C⊥ 1 : Fn q) implies the second

statement.

Duals of Graph Codes

Theorem (G, C1 : C2) = (G, C1 : Fn

q) ∩ (G, Fn q : C2)

(G, C1 : C2)⊥ = (G, C⊥

1 : Fn q) + (G, Fn q : C⊥ 2 ).

This theorem also improves the usual bound on the dimension of Graph codes. The usual bound counts checks given by (G, C⊥

1 : Fn q) and

(G, Fn

q : C⊥ 2 ). We improve it by counting the dependent

checks in common: (G, C⊥

1 : C⊥ 2 ).

Properties of Graph Codes

Let G be an n–regular, bipartite graph. Let C be a [n, k, d]

• code. Let γG be the ratio of the second largest and largest

eigenvalues of (the adjacency matrix of) G, then d(G, C : C) ≥ |E| d

n

d n −γG

1−γG

(G, C : C) can be decoded in linear time.

A Problem

But determining the dimension of Graph Based codes remains difficult.

Affine Variety Codes

Let V = {P1, P2, . . . , Pn} be a finite set of points over a finite field. Let I = I(V) be the ideal of polynomials which vanish at these points. Let L be a Fq-linear subspace of Fq[x1, x2, . . . , xm]/I We define the affine variety code as C(I, L) := {(f(P1), f(P2), . . . , f(Pm)) | f ∈ L}.

Reed–Solomon Codes

We define Reed–Solomon codes as follows. Let Ik := {0, 1, . . . , k − 1} ⊆ Zq−1. Let M(IX) := {ti | i ∈ IX}. The code C := C(Fq, M(IX)Fq) is the Reed–Solomon code RSq(q, k). We identify the positions of C with the elements of Fq.

Our Graph

We construct codes over the following graph Γ. V1 = (x, y) ∈ Fq × Fq V2 = (a, b) ∈ Fq × Fq ((x, y), (a, b)) ∈ E(Γ) if and only if ax + b = y. Γ is a q regular bipartite subgraph of the Affine Plane point line incidence graph.

Algebraic Formulation of the Edge Set

Consider the points (x, y, a, b) ∈ F4

q which satisfy:

ax = y − b. We study the ideal I(Γ) := ax + b − y, aq − a, y q − y, xq − x, bq − b instead. This is the ideal corresponding to the points (x, y, a, b) ∈ F4

q .

Algebraic Formulation of Graph Codes

We use the following labeling: φ(x,y)((x, y, a, b)) = a φ(a,b)((x, y, a, b)) = x

Some special monomial sets

∆(1)

1

:= {X i1Y i2Bj2 | i2 < q and i1 + j2 < q}. (4) ∆(2)

1

:= {Y i2Aj1Bj2 | i2 < q and j1 + j2 < q}. (5) ∆1 has deg X, deg A < q − deg B and deg Y < q. ∆(1)

2

:= {X i1Y i2Bj2 | j2 < q and i1 + i2 < q}. (6) ∆(2)

2

:= {Y i2Aj1Bj2 | j2 < q and i2 + j1 < q}. (7) ∆2 has deg X, deg A < q − deg Y and deg B < q. ∆1 = ∆(1)

1

∪ ∆(2)

1

and ∆2 = ∆(1)

2

∪ ∆(2)

2

(8)

Some special monomial sets

∆(1)

1

:= {X i1Y i2Bj2 | i2 < q and i1 + j2 < q}. (4) ∆(2)

1

:= {Y i2Aj1Bj2 | i2 < q and j1 + j2 < q}. (5) ∆1 has deg X, deg A < q − deg B and deg Y < q. ∆(1)

2

:= {X i1Y i2Bj2 | j2 < q and i1 + i2 < q}. (6) ∆(2)

2

:= {Y i2Aj1Bj2 | j2 < q and i2 + j1 < q}. (7) ∆2 has deg X, deg A < q − deg Y and deg B < q. ∆1 = ∆(1)

1

∪ ∆(2)

1

and ∆2 = ∆(1)

2

∪ ∆(2)

2

(8)

Some special monomial sets

∆(1)

1

:= {X i1Y i2Bj2 | i2 < q and i1 + j2 < q}. (4) ∆(2)

1

:= {Y i2Aj1Bj2 | i2 < q and j1 + j2 < q}. (5) ∆1 has deg X, deg A < q − deg B and deg Y < q. ∆(1)

2

:= {X i1Y i2Bj2 | j2 < q and i1 + i2 < q}. (6) ∆(2)

2

:= {Y i2Aj1Bj2 | j2 < q and i2 + j1 < q}. (7) ∆2 has deg X, deg A < q − deg Y and deg B < q. ∆1 = ∆(1)

1

∪ ∆(2)

1

and ∆2 = ∆(1)

2

∪ ∆(2)

2

(8)

Lemma ∆1 and ∆2 are normal basis for I(Γ). Computing a Gröbner basis for I(Γ) under degree graded reverse lexicographical order with X > Y > A > B will add the polynomials of the form X i(Y − B)q−i − Aq−1−i(Y − B) and Ai(Y − B)q−i − X q−1−i(Y − B) to the basis. ∆1 consists of all monomials which are not divisible by any of the leading terms.

Properties of Affine Variety Codes

Theorem Let F[ ¯ X] be a polynomial ring, and let I be an ideal of F[ ¯ X] such that we may construct Affine Variety codes with

• I. Let L, M be F-subspaces of normal bases for

R = F[ ¯ X]/I, then C(I, L) ∩ C(I, M) = C(I, L ∩R M) dim L ∩R M = dim L ∩ R + dim(I ∩ (L ∪ M))

Properties of Affine Variety Codes

Theorem dim L ∩R M = dim L ∩ M + dim(I ∩ (L ∪ M)) Let f ∈ L. Let rem(f) be the remainder of f in the normal basis containing M. The map φ : L ∩R M → I ∩ (L ∪ M), f → f − rem(f) is a surjective mapping from L ∩R M to I(Γ) ∩ (L ∪ M)). Its kernel is L ∩ M.

Some more special monomial sets

∆1(k) := {X i1Y i2Aj1Bj2 ∈ ∆1 | j1 + j2 < k}. (9) deg X + deg B < q, deg A + deg B < k ∆2(k) := {X i1Y i2Aj1Bj2 ∈ ∆2 | i1 + i2 < k}. (10) deg A + deg Y < q, deg X + deg Y < k

Algebraic Formulation of Graph Codes (cont.)

For the labeling we have chosen, we have the following equalities: (Γ, RSq(q, k) : Fq

q) = C(I(Γ), ∆1(k))

(Γ, Fq

q : RSq(q, k)) = C(I(Γ), ∆2(k))

Relation between Graph Codes and Affine Variety Codes

Theorem (Γ, RSq(q, k) : Fq

q) = C(I(Γ), ∆1(k))

Let f(X, Y, A, B) ∈ ∆1(k). Let X = x and Y = y. The univariate polynomial f(x, y, A, y − Ax) is a polynomial of degree at most k − 1. Therefore its evaluation is a codeword in RSq(q, k). Equality follows from counting the dimension of both spaces.

Relation between Graph Codes and Affine Variety Codes

Theorem (Γ, RSq(q, k) : Fq

q) = C(I(Γ), ∆1(k))

Let f(X, Y, A, B) ∈ ∆1(k). Let X = x and Y = y. The univariate polynomial f(x, y, A, y − Ax) is a polynomial of degree at most k − 1. Therefore its evaluation is a codeword in RSq(q, k). Equality follows from counting the dimension of both spaces.

Relation between Graph Codes and Affine Variety Codes (cont.)

Similarily, (Γ, Fq

q : RSq(q, k)) = C(I(Γ), ∆2(k))

Therefore (Γ, RSq(q, k) : RSq(q, k)) = C(I(Γ), ∆1(k)) ∩ C(I(Γ), ∆2(k)) = C(I(Γ), ∆1(k) ∩ ∆2(k))

Relation between Graph Codes and Affine Variety Codes (cont.)

Similarily, (Γ, Fq

q : RSq(q, k)) = C(I(Γ), ∆2(k))

Therefore (Γ, RSq(q, k) : RSq(q, k)) = C(I(Γ), ∆1(k)) ∩ C(I(Γ), ∆2(k)) = C(I(Γ), ∆1(k) ∩ ∆2(k))

Computation of the dimension of the code C(I(Γ), ∆1(k) ∩ ∆2(k))

If k ≤ q/2 then the set-wise intersection ∆1(k) ∩ ∆2(k) gives k 3. ∆1 and ∆2 we chosen as normal basis for Fq[X, Y, A, B]/I(Γ). Moreover, the Gröbner basis which give these normal basis are (almost) identical. There exists f ∈ I(Γ) ∩ (∆1(k) ∪ ∆2(k)) if and only if the homogeneous multiples of (Y − B)q−k have no nonzero middle terms. Impossible for q even or prime. The setwise intersection is the true dimension (for q even or prime.).

Computation of the dimension of the code C(I(Γ), ∆1(k) ∩ ∆2(k))

If k ≤ q/2 then the set-wise intersection ∆1(k) ∩ ∆2(k) gives k 3. ∆1 and ∆2 we chosen as normal basis for Fq[X, Y, A, B]/I(Γ). Moreover, the Gröbner basis which give these normal basis are (almost) identical. There exists f ∈ I(Γ) ∩ (∆1(k) ∪ ∆2(k)) if and only if the homogeneous multiples of (Y − B)q−k have no nonzero middle terms. Impossible for q even or prime. The setwise intersection is the true dimension (for q even or prime.).

Computation of the dimension of the code C(I(Γ), ∆1(k) ∩ ∆2(k))

If k ≤ q/2 then the set-wise intersection ∆1(k) ∩ ∆2(k) gives k 3. ∆1 and ∆2 we chosen as normal basis for Fq[X, Y, A, B]/I(Γ). Moreover, the Gröbner basis which give these normal basis are (almost) identical. There exists f ∈ I(Γ) ∩ (∆1(k) ∪ ∆2(k)) if and only if the homogeneous multiples of (Y − B)q−k have no nonzero middle terms. Impossible for q even or prime. The setwise intersection is the true dimension (for q even or prime.).

Computation of the dimension of the code C(I(Γ), ∆1(k) ∩ ∆2(k))

If k ≤ q/2 then the set-wise intersection ∆1(k) ∩ ∆2(k) gives k 3. ∆1 and ∆2 we chosen as normal basis for Fq[X, Y, A, B]/I(Γ). Moreover, the Gröbner basis which give these normal basis are (almost) identical. There exists f ∈ I(Γ) ∩ (∆1(k) ∪ ∆2(k)) if and only if the homogeneous multiples of (Y − B)q−k have no nonzero middle terms. Impossible for q even or prime. The setwise intersection is the true dimension (for q even or prime.).

Computation of the dimension of the code C(I(Γ), ∆1(k) ∩ ∆2(k))

If k ≤ q/2 then the set-wise intersection ∆1(k) ∩ ∆2(k) gives k 3. ∆1 and ∆2 we chosen as normal basis for Fq[X, Y, A, B]/I(Γ). Moreover, the Gröbner basis which give these normal basis are (almost) identical. There exists f ∈ I(Γ) ∩ (∆1(k) ∪ ∆2(k)) if and only if the homogeneous multiples of (Y − B)q−k have no nonzero middle terms. Impossible for q even or prime. The setwise intersection is the true dimension (for q even or prime.).

Codewords

The codewords in the case k ≤ q

2 can now be obtained as

evaluations of polynomials of degree lower than k in (a, b) and (x, y) that do not have y − ax − b as a factor.

Codewords

To obtain the set of polynomials that are evaluated to codewords when k > q

2 we use as above Gröbner basis

techniques . In characteristic 2 we get a Gröbner basis consisting of the original 5 polynomials and yaq−1 + y + baq−1 + b y 2aq−2 + b2aq−2 + bx + xy . . . y q−1a + y q−2ba + . . . + bq−1a + bxq−2 + xq−2y xq−1y + bxq−1 + y + b xq−2y 2 + xq−2b2 + ay + ab . . .

Codewords

Among the monomials in the footprint, all polynomials of degree less than k in (a, b) and (x, y) evaluate to

• codewords. In addition the weighted degree basis

provides the monomials which have degree < k in (x, y), but higher degree in (a, b). By reversing the total order to degree(a, b) >> degree(x, y), we can reduce these monomials to polynomials with the lowest possible degree in (a, b) and find the space that has low degree in both

• representations. Thus these additional functions have two

equivalent representations, one with degree < k in (a, b) and another with degree < k in (x, y). The procedure is illustrated in the example.

Codewords

Among the monomials in the footprint, all polynomials of degree less than k in (a, b) and (x, y) evaluate to

• codewords. In addition the weighted degree basis

provides the monomials which have degree < k in (x, y), but higher degree in (a, b). By reversing the total order to degree(a, b) >> degree(x, y), we can reduce these monomials to polynomials with the lowest possible degree in (a, b) and find the space that has low degree in both

• representations. Thus these additional functions have two

equivalent representations, one with degree < k in (a, b) and another with degree < k in (x, y). The procedure is illustrated in the example.

Codewords

Among the monomials in the footprint, all polynomials of degree less than k in (a, b) and (x, y) evaluate to

• codewords. In addition the weighted degree basis

provides the monomials which have degree < k in (x, y), but higher degree in (a, b). By reversing the total order to degree(a, b) >> degree(x, y), we can reduce these monomials to polynomials with the lowest possible degree in (a, b) and find the space that has low degree in both

• representations. Thus these additional functions have two

equivalent representations, one with degree < k in (a, b) and another with degree < k in (x, y). The procedure is illustrated in the example.

Example For q = 16, N = 212, and the dimensions of the codes for k = 1 to 15 are 1, 8, 27, 64, 125, 216, 343, 512, 855, 1240, 1661, 2112, 2587, 3080, 3585. Part of the basis for the code with k = 12 is obtained by evaluating all monomials of degree < 12 in both (x, y) and (a, b). It follows from the lemma that there 123 = 1728 such monomials, and 384 additional basis functions are needed.

Example cont.

Considering those polynomials in the Gröbner basis with degree(x, y) > 11, we find y 12a4 + x11y = y 8b4a4 + y 4b8a4 + ba11 y 13a3 + y 12ba3 + x12y = y 9b4a3 + y 8b5a3 + . . . + b13a3 + bx12 y 14a2 + y 12b2a2 + x13y = y 10b4a2 + y 8b6a2 + . . . + b14a2 + bx13 y 15a + y 14ba + y 13b2a + y 12b3a + x14y = y 11b4a + y 10b5a + . . . + b15a + bx14 We manipulate these to give the remaining 384 basis functions.

Codes on projective planes

In the projective plane there are m = q2 + q + 1 points and lines The component codes are n = q + 1 doubly extended RS codes Points have coordinates (x : y : z) and lines (a : b : c), where a point is on a line if axr + by r + czr = 0 where q = r 2 Component codewords are evaluations of homogeneous polynomials in two variables with degree k − 1 Denote the code Cp(q, k).

Partitioning of the plane into disjoint subplanes

The projective plane over Fq can be partitioned into q − r + 1 disjoint subplanes over the subfield Fr Example: The F4 plane with 21 points and lines can be partioned into the binary plane ( 7 points) and two shifts of this plane. Each line has r + 1 points in one subplane and one point in each of the other subplanes ( similarly for line through a given point). The supplane gives a subgraph with degree r + 1, and by combining j copies, we get a subgraph of degree r + j. These subgraphs are the smallest possible for a graph with this degree and second largest eigenvalue r.

Partitioning of the plane into disjoint subplanes

The projective plane over Fq can be partitioned into q − r + 1 disjoint subplanes over the subfield Fr Example: The F4 plane with 21 points and lines can be partioned into the binary plane ( 7 points) and two shifts of this plane. Each line has r + 1 points in one subplane and one point in each of the other subplanes ( similarly for line through a given point). The supplane gives a subgraph with degree r + 1, and by combining j copies, we get a subgraph of degree r + j. These subgraphs are the smallest possible for a graph with this degree and second largest eigenvalue r.

Minimum distance

The minimal subgraph of degree d gives a lower bound on the minimum distance of the graph codes as D ≥ md(d − r)/(n − r) Evaluation of ax + by + cz gives 0 on the Fr subplane, but not on the shifted versions. Similarly there are degree 1 polynomials that evaluate to 0 on each of the other subplanes. By taking a product of degree k − 1 we get a codeword weight equal to the lower bound. Thus for d > r, the bound is the exact value.

Dimension

The dimension may be upper bounded by the size of a forcing set. First select all edges on a subgraph of degree k ( as describes previously). Thus all symbols in these subplanes are known. It follows from the construction that k − r symbols are known in each of the remaining component codes. The remaining dimension is upper bounded by the dimension of C(q, r), and for low degrees, the dimension can be found directly by counting polynomials. The upper bound becomes K ≤ mk(k − r)/(n − r) + dim(r)

Dimension

For degrees less than r, all monomials of a given degree are independent. For degree ≥ r we must subtract polynomials that are multiples of the two versions of the line equation. For degree > r we then add multiples of the product

• f these two polynomials ( they were counted twice

above). The result is the same as the lower bound, and we have the exact value of the dimension: K = mk(k − r)/(n − r) + [r(r + 1)/2]2 When n is not too small, the result is close to the rate

• f the product code.

Codes on Euclidean Planes

In the Euclidean plane there are m = q2 points and lines ( no vertical lines). The component codes are n = q extended RS codes Points have coordinates (x, y), lines (a, b), where a point is on a line if axr + b − y r = 0, q = r 2. Component codewords are evaluations of polynomials in one variable with degree < k.

Special Polynomial

From the line equations axr + b − y r = 0 and arx + br − y = 0 we get ar−1b + brxr−1 = ar−1y r − xr−1y Here one side has degree r in a, b but degree r − 1 in x, y and the degrees are opposite on the other side. Thus the special polynomial evaluates to a codeword with k = r. This is the basic tool in the modifications required for the proofs of the dimension and the minimum distance of the Euclidean graph codes.

Partitioning the plane into subplanes

The Euclidean plane over Fq can be partitioned into q disjoint subplanes over the subfield Fr. Example: The F4 plane with16 points and lines can be partitioned into the binary plane ( 4 points) and three shifts of this plane. Each line has r points in one subplane and one point in each of q − r other subplanes. However it has no points on r − 1 subplanes ( parallel to the first subplane). By combining one subplane from each of the r sets of parallel subplanes we get a subgraph of degree 2r − 1. Similarly we can take j from each set. These subgraphs are the smallest possible for a graph with this degree and second largest eigenvalue r.

Partitioning the plane into subplanes

The Euclidean plane over Fq can be partitioned into q disjoint subplanes over the subfield Fr. Example: The F4 plane with16 points and lines can be partitioned into the binary plane ( 4 points) and three shifts of this plane. Each line has r points in one subplane and one point in each of q − r other subplanes. However it has no points on r − 1 subplanes ( parallel to the first subplane). By combining one subplane from each of the r sets of parallel subplanes we get a subgraph of degree 2r − 1. Similarly we can take j from each set. These subgraphs are the smallest possible for a graph with this degree and second largest eigenvalue r.

Minimum distance

The minimal subgraph of degree d gives a lower bound on the minimum distance of the graph codes as D ≥ md(d − r)/(n − r) Evaluation of ax + b − y gives 0 on the Fr. subplane, but not on the shifted versions. Similarly there are

• ther degree 1 polynomials that evaluate to 0 on

each of the other subplanes. By taking a product of degree k we get a word with weight equal to the lower bound. However, it is not clear that it is a codeword in the code under consideration. Start out with k = r, d = q − r + 1. Of every r parallel subplanes, r − 1 are nonzero, and r should be zero. We use the special polynomial to show that the evaluation is a codeword.

Minimum distance

The minimal subgraph of degree d gives a lower bound on the minimum distance of the graph codes as D ≥ md(d − r)/(n − r) Evaluation of ax + b − y gives 0 on the Fr. subplane, but not on the shifted versions. Similarly there are

• ther degree 1 polynomials that evaluate to 0 on

each of the other subplanes. By taking a product of degree k we get a word with weight equal to the lower bound. However, it is not clear that it is a codeword in the code under consideration. Start out with k = r, d = q − r + 1. Of every r parallel subplanes, r − 1 are nonzero, and r should be zero. We use the special polynomial to show that the evaluation is a codeword.

dimension

The dimension may be upper bounded by the size of a forcing set , starting with a degree k subgraph as before. The upper bound becomes K ≤ mk(k − r)/(n − r) + dim(r) Counting polynomials of degree less than k gives the same result as counting homogeneous polynomials

• f degree k − 1

However, we have to add multiples of the special polynomial, and later powers of this polynomial. In this way we get the same expression as for the upper bound, and thus we have the exact dimension.

dimension

The dimension may be upper bounded by the size of a forcing set , starting with a degree k subgraph as before. The upper bound becomes K ≤ mk(k − r)/(n − r) + dim(r) Counting polynomials of degree less than k gives the same result as counting homogeneous polynomials

• f degree k − 1

However, we have to add multiples of the special polynomial, and later powers of this polynomial. In this way we get the same expression as for the upper bound, and thus we have the exact dimension.

Example

For q = 16 some parameter values (k = 7 and 10) are : Projective codes (4641, 541, 1617) and (4641, 1360, 672) Euclidean codes (4096, 549, 1280) and (4096, 1381, 448) The field is just large enough to demonstrate the results in this presentation, but to get interesting codes ( higher rates in particular), q must be at least 256, more likely 1024

Performance

We use iterative decoding and assume that decoding of the component Reed-Solomon codes either corrects the errors or fails to produce a result. In the latter case the received word is left unchanged.

Error Pattern

The error pattern can be described by a graph which is

• btained from the original bipartite graph by including only

branches containing errors. Iterative decoding is then described as a process of removing a node with at most t branches and any branches connecting to the node. It is well-known that the process terminates with an empty graph or with a subgraph where all nodes have degree at least t + 1.

Error Pattern

Any error pattern of weight less than D/4 is decoded in this way. This result is similar to the decoding of product codes by rows and columns, but it should be noted that for graph codes, the minimum distance increases linearly with the code length. If a set of nodes is not decoded, it is possible to erase the corresponding symbols and increase the number or errors that can be corrected to at least D/2.

Performance

The performance of the graph codes under iteration of the decoding of the component codes can be analyzed using methods of random graphs. It follows that the code will be successfully decoded with high probability even if the average number of errors in each component RS code is slightly larger than (q − k)/2. Thus in most cases m(q−k)

2

errors are decoded.

Performance

For q = 16 and k = 12, the rate of the code from the Euclidean plane is 0.5156. The lower bound on the minimum distance is 105, but we expect to be able to correct 512 symbol errors in most cases. For correcting binary errors we could represent each symbol as 5 bits with a parity symbol. The lower bound on the minimum distance is 210 in this case, but we expect to correct about 1000 binary errors.

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