Affine variety codes are better than their reputation Olav Geil - - PowerPoint PPT Presentation

Affine variety codes are better than their reputation

Olav Geil Aalborg University (joint with Stefano Martin) International conference on Algebraic Geometry and Coding Theory IIT Bombay December 2013

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

Affine variety codes

I ⊆ Fq[X1, . . . , Xm] Iq = I + X q

1 − X1, . . . , X q m − Xm.

{P1, . . . , Pn} = VFq(Iq), {N1 + Iq, . . . , Nn + Iq} a basis for Fq[X1, . . . , Xm]/Iq. We get a basis for Fn

q:

{ b1 =

• N1(P1), . . . , N1(Pn)
• , . . . ,

bn =

• Nn(P1), . . . , Nn(Pn)
• }

Definition Consider L ⊆ {1, . . . , n}. C(I, L) = SpanFq{ bi | i ∈ L} C ⊥(I, L) =

• C(I, L)

⊥. Theorem C is a linear code ⇔ C is an affine variety code.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

Affine variety codes

I ⊆ Fq[X1, . . . , Xm] Iq = I + X q

1 − X1, . . . , X q m − Xm.

{P1, . . . , Pn} = VFq(Iq), {N1 + Iq, . . . , Nn + Iq} a basis for Fq[X1, . . . , Xm]/Iq. We get a basis for Fn

q:

{ b1 =

• N1(P1), . . . , N1(Pn)
• , . . . ,

bn =

• Nn(P1), . . . , Nn(Pn)
• }

Definition Consider L ⊆ {1, . . . , n}. C(I, L) = SpanFq{ bi | i ∈ L} C ⊥(I, L) =

• C(I, L)

⊥. Theorem C is a linear code ⇔ C is an affine variety code.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

One-point AG codes

Theorem If Q is a rational place then ∪∞

s=0L(sQ) ≃ Fq[X1, . . . , Xm]/I where

I satisfies the order domain conditions. Theorem A map h : Fq[X1, . . . , Xm]/I → Fn

q such that

h is Fq-linear, h(f ) = (c1, . . . , cn) and h(g) = (d1, . . . , dn) ⇒ h(fg) = (c1d1, . . . , cndn) is of the form h(f = F + I) = (F(P1), . . . , F(Pn)), where P1, . . . , Pn are affine points.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

Our work

Most known affine variety codes are one-point AG codes in disguise. We introduce a much broader class of affine variety codes. We

generalise the Feng-Rao-bound/order-bound for dual codes (also simply known as the Feng-Rao-bound/order-bound). Our method builds on work by Salazar et al. generalise the Feng-Rao-bound/order-bound for primary codes (sometimes called the Andersen–G bound),

We treat affine variety codes and general linear codes. We treat minimum distance and generalised Hamming weights.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

Our work

Most known affine variety codes are one-point AG codes in disguise. We introduce a much broader class of affine variety codes. We

generalise the Feng-Rao-bound/order-bound for dual codes (also simply known as the Feng-Rao-bound/order-bound). Our method builds on work by Salazar et al. generalise the Feng-Rao-bound/order-bound for primary codes (sometimes called the Andersen–G bound),

We treat affine variety codes and general linear codes. We treat minimum distance and generalised Hamming weights.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

Our work

Most known affine variety codes are one-point AG codes in disguise. We introduce a much broader class of affine variety codes. We

generalise the Feng-Rao-bound/order-bound for dual codes (also simply known as the Feng-Rao-bound/order-bound). Our method builds on work by Salazar et al. generalise the Feng-Rao-bound/order-bound for primary codes (sometimes called the Andersen–G bound),

We treat affine variety codes and general linear codes. We treat minimum distance and generalised Hamming weights.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

Our work

Most known affine variety codes are one-point AG codes in disguise. We introduce a much broader class of affine variety codes. We

generalise the Feng-Rao-bound/order-bound for dual codes (also simply known as the Feng-Rao-bound/order-bound). Our method builds on work by Salazar et al. generalise the Feng-Rao-bound/order-bound for primary codes (sometimes called the Andersen–G bound),

We treat affine variety codes and general linear codes. We treat minimum distance and generalised Hamming weights.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

The footprint bound

Definition Given an ideal J ⊆ k[X1, . . . , Xm] and a monomial ordering ≺ then ∆≺(J) = {M is a monomial | M / ∈ lm(J)} Theorem (The footprint bound:) If J ⊆ k[X1, . . . , Xm] is radical and zero-dimensional and if k is a perfect field then #V(J) = #∆≺(J).

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

The footprint bound and other bounds

Theorem (The footprint bound:) If J ⊆ k[X1, . . . , Xm] is radical and zero-dimensional and if k is a perfect field then #V(J) = #∆≺(J). For primary order domain codes (one-point AG codes, generalised Reed-Muller codes, etc.) the order bound is a consequence of the footprint bound. Our new bound for primary codes relies on the footprint bound. Our new bound for dual codes uses Feng-Rao arguments, and the connection to the primary bound is not completely clear.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

The footprint bound and other bounds

Theorem (The footprint bound:) If J ⊆ k[X1, . . . , Xm] is radical and zero-dimensional and if k is a perfect field then #V(J) = #∆≺(J). For primary order domain codes (one-point AG codes, generalised Reed-Muller codes, etc.) the order bound is a consequence of the footprint bound. Our new bound for primary codes relies on the footprint bound. Our new bound for dual codes uses Feng-Rao arguments, and the connection to the primary bound is not completely clear.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

The rest of this talk

Our bound for primary codes can easily be explained for affine variety codes. To explain our bound for dual codes we need to go to linear code level. Agenda: We start by studying the order domain conditions and primary codes. Then we throw away half of the order domain conditions and consider primary codes. We go to linear code level and explain the bound for dual codes. We present numerical data for both primary and dual codes.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

The rest of this talk

Our bound for primary codes can easily be explained for affine variety codes. To explain our bound for dual codes we need to go to linear code level. Agenda: We start by studying the order domain conditions and primary codes. Then we throw away half of the order domain conditions and consider primary codes. We go to linear code level and explain the bound for dual codes. We present numerical data for both primary and dual codes.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

Hermitian code

I = X 2 + X − Y 3 ⊆ F4[X, Y ], Iq = I + X 4 − X, Y 4 − Y . A weighted degree lexicographic ordering From the weight function w(X iY j) = 3i + 2j we define the monomial ordering ≺w by N ≺w M if either w(N) < w(M),

• r w(N) = w(M) but degX(N) < degX(M).

{P1, . . . , P8} = V(Iq). Consider c = (F(P1), . . . , F(P8)). wH( c) = 8 − # common zeros between F and Iq = #

• ∆≺w (Iq)\∆≺w (Iq + F)
• =

#{M ∈ ∆≺w (Iq) | M ∈ lm(Iq + F)}.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

Hermitian code - cont.

Consider c = (F(P1), . . . , F(P8)), say F = a1 + a2Y + X wH( c) = #{M ∈ ∆≺w (Iq) | M ∈ lm(Iq + F)}. Y 3 XY 3 6 9 Y 2 XY 2 4 7 Y XY 2 5 1 X 3 X = lm(F), XY = lm(YF), XY 2 = lm(Y 2F), XY 3 = lm(Y 3F), Y 3 = lm(XF −(X 2 +X −Y 3)) In conclusion, wH( c) ≥ 5. We could also have counted the numbers in {0, 2, 3, 4, 5, 6, 7, 9} which are being hit by w(lm(F)) = 3. This is due to X 2 + X − Y 3 having two monomials of the highest weight and all monomials in ∆≺w (I) being of different weight.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

Hermitian code - cont.

Consider c = (F(P1), . . . , F(P8)), say F = a1 + a2Y + X wH( c) = #{M ∈ ∆≺w (Iq) | M ∈ lm(Iq + F)}. Y 3 XY 3 6 9 Y 2 XY 2 4 7 Y XY 2 5 1 X 3 X = lm(F), XY = lm(YF), XY 2 = lm(Y 2F), XY 3 = lm(Y 3F), Y 3 = lm(XF −(X 2 +X −Y 3)) In conclusion, wH( c) ≥ 5. We could also have counted the numbers in {0, 2, 3, 4, 5, 6, 7, 9} which are being hit by w(lm(F)) = 3. This is due to X 2 + X − Y 3 having two monomials of the highest weight and all monomials in ∆≺w (I) being of different weight.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

The order domain conditions

Definition Consider an ideal J ⊆ k[X1, . . . , Xm] where k is a field. Let a weighted degree ordering ≺w be given. Assume that J possesses a Gr¨

• bner basis F with respect to ≺w such that:

(C1) Any F ∈ F has exactly two monomials of highest weight. (C2) No two monomials in ∆≺w (J) are of the same weight. Then we say that J and ≺w satisfy the order domain conditions. The Feng-Rao bounds do not work well when the order domain conditions are not satisfied. We throw away condition (C2) and introduce a method that works well for the corresponding codes.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

The order domain conditions

Definition Consider an ideal J ⊆ k[X1, . . . , Xm] where k is a field. Let a weighted degree ordering ≺w be given. Assume that J possesses a Gr¨

• bner basis F with respect to ≺w such that:

(C1) Any F ∈ F has exactly two monomials of highest weight. (C2) No two monomials in ∆≺w (J) are of the same weight. Then we say that J and ≺w satisfy the order domain conditions. The Feng-Rao bounds do not work well when the order domain conditions are not satisfied. We throw away condition (C2) and introduce a method that works well for the corresponding codes.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

An affine variety code over F8.

I = (X 4 + X 2 + X) − (Y 6 + Y 5 + Y 3) ⊆ F8[X, Y ]. Iq = I + X 8 − X, Y 8 − Y . Define ≺w on the basis of w(X iY j) = 3i + 2j. Y 7 XY 7 X 2Y 7 X 3Y 7 Y 6 XY 6 X 2Y 6 X 3Y 6 Y 5 XY 5 X 2Y 5 X 3Y 5 Y 4 XY 4 X 2Y 4 X 3Y 4 Y 3 XY 3 X 2Y 3 X 3Y 3 Y 2 XY 2 X 2Y 2 X 3Y 2 Y XY X 2Y X 3Y 1 X X 2 X 3 ∆≺w (Iq) 14 17 20 23 12 15 18 21 10 13 16 19 8 11 14 17 6 9 12 15 4 7 10 13 2 5 8 11 3 6 9 Corresponding weights

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

An affine variety code over F8 - cont.

14 17 20 23 12 15 18 21 10 13 16 19 8 11 14 17 6 9 12 15 4 7 10 13 2 5 8 11 3 6 9 V(Iq) = {P1, . . . , P32}

• c = (F(P1), . . . , F(P32))

where F = a1 + a2Y + a3X + a4Y 2 +a5XY + a6Y 3 + a7X 2 + a8XY 2 +a9Y 4 + a10X 2Y + a11XY 3 + X 3 Observe that w(XY 3) = w(X 3) = 9. Hence, we must be careful.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

An affine variety code over F8 - cont.

F = a1 + a2Y + a3X + a4Y 2 + a5XY + a6Y 3 + a7X 2 + a8XY 2 + a9Y 4 + a10X 2Y + a11XY 3 + X 3. 14 17 20 23 12 15 18 21 10 13 16 19 8 11 14 17 6 9 12 15 4 7 10 13 2 5 8 11 3 6 9 Case 1: a11 = 0 lm

• XF −
• (X 4 + X 2 + X) − (Y 6 +

Y 5 + Y 3)

• = Y 6 and therefore we

find not only X 3, X 3Y ,X 3Y 2,X 3Y 3, X 3Y 4, X 3Y 5, X 3Y 6, X 3Y 7 but also Y 6, XY 6, X 2Y 6, Y 7, XY 7, X 2Y 7 as leading monomials. Remember: wH( c) = #{M ∈ ∆≺w (Iq) | M ∈ lm(Iq + F)}.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

An affine variety code over F8 - cont.

F = a1 + a2Y + a3X + a4Y 2 + a5XY + a6Y 3 + a7X 2 + a8XY 2 + a9Y 4 + a10X 2Y + a11XY 3 + X 3. 14 17 20 23 12 15 18 21 10 13 16 19 8 11 14 17 6 9 12 15 4 7 10 13 2 5 8 11 3 6 9 Case 2: a11 = 0 lm

• XF −
• (X 4+X 2+X)−(Y 6+Y 5+

Y 3)

• = X 2Y 3 and therefore we

find not only X 3, X 3Y ,X 3Y 2,X 3Y 3, X 3Y 4, X 3Y 5, X 3Y 6, X 3Y 7 but also X 2Y 3, X 2Y 4, X 2Y 5, X 2Y 6, X 2Y 7 as leading monomials. Case 1 gave wH( c) ≥ 14 and Case 2 gave wH( c) ≥ 13. Hence, wH( c) ≥ 13. (The Feng-Rao bound (WB) gives wH( c) ≥ 8)

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

An affine variety code over F8 - cont.

F = a1 + a2Y + a3X + a4Y 2 + a5XY + a6Y 3 + a7X 2 + a8XY 2 + a9Y 4 + a10X 2Y + a11XY 3 + X 3. 14 17 20 23 12 15 18 21 10 13 16 19 8 11 14 17 6 9 12 15 4 7 10 13 2 5 8 11 3 6 9 Case 2: a11 = 0 lm

• XF −
• (X 4+X 2+X)−(Y 6+Y 5+

Y 3)

• = X 2Y 3 and therefore we

find not only X 3, X 3Y ,X 3Y 2,X 3Y 3, X 3Y 4, X 3Y 5, X 3Y 6, X 3Y 7 but also X 2Y 3, X 2Y 4, X 2Y 5, X 2Y 6, X 2Y 7 as leading monomials. Case 1 gave wH( c) ≥ 14 and Case 2 gave wH( c) ≥ 13. Hence, wH( c) ≥ 13. (The Feng-Rao bound (WB) gives wH( c) ≥ 8)

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

Terminology for general linear code

Feng-Rao introduced the concept of well-behaving pairs (WB), Miura the concept of weakly well-behaving pairs (WWB), G–Thommesen the concept of one-way well-behaving pairs (OWB). OWB ⇐ WWB ⇐ WB Therefore OWB gives the strongest bounds. OWB becomes crucial when we skip the second order domain condition.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

A basis for Fn

q

I = (X 4 + X 2 + X) − (Y 6 + Y 5 + Y 3) ⊆ F8[X, Y ]. Iq = I + X 8 − X, Y 8 − Y . ∆≺w (Iq) = {N1, . . . , N32}, with N1 ≺w · · · ≺w N32. 14 17 20 23 12 15 18 21 10 13 16 19 8 11 14 17 6 9 12 15 4 7 10 13 2 5 8 11 3 6 9 b21 b26 b30 b32 b17 b23 b28 b31 b13 b19 b25 b29 b9 b15 b22 b27 b6 b11 b18 b24 b4 b8 b14 b20 b2 b5 b10 b16 b1 b3 b7 b12

• bi =
• Ni(P1), . . . , Ni(P32)
• .

B = { b1, · · · , b32} a basis for F32

8 .

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

OWB

Let G be a Gr¨

• bner basis for Iq and ∆≺(Iq) = {N1, . . . , Nn}.

Nice situation: lm t

i=1 aiNi

• Nj rem G
• = lm
• NiNj rem G
• = Nl

for all choices of a1, . . . , at such that at = 0. Corresponds to (i, j) being OWB. Our method treats also less nice situations.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

OWB

Let G be a Gr¨

• bner basis for Iq and ∆≺(Iq) = {N1, . . . , Nn}.

Nice situation: lm t

i=1 aiNi

• Nj rem G
• = lm
• NiNj rem G
• = Nl

for all choices of a1, . . . , at such that at = 0. Corresponds to (i, j) being OWB. Our method treats also less nice situations.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

Feng-Rao bound for primary codes

{ b1, . . . , bn} a basis for Fn

q.

For c = t

i=1 ai

bi, at = 0 define ¯ ρ( c) = t.

• α ∗

β = (α1β1, . . . , αnβn). If ¯ ρ t

i=1 ai

bi

• ∗

bj

• = ¯

ρ( bi ∗ bj) = l for all choices of a1, . . . , at such that at = 0 then (i, j) is OWB. Theorem (The Feng-Rao bound for primary codes) If at = 0 then wH t

i=1 ai

bi

• is at least equal to the number of

l ∈ {1, . . . , n} for which a j exists with (i, j) OWB and ¯ ρ( bi ∗ bj) = l. Our results are stronger.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

Feng-Rao bound for primary codes

{ b1, . . . , bn} a basis for Fn

q.

For c = t

i=1 ai

bi, at = 0 define ¯ ρ( c) = t.

• α ∗

β = (α1β1, . . . , αnβn). If ¯ ρ t

i=1 ai

bi

• ∗

bj

• = ¯

ρ( bi ∗ bj) = l for all choices of a1, . . . , at such that at = 0 then (i, j) is OWB. Theorem (The Feng-Rao bound for primary codes) If at = 0 then wH t

i=1 ai

bi

• is at least equal to the number of

l ∈ {1, . . . , n} for which a j exists with (i, j) OWB and ¯ ρ( bi ∗ bj) = l. Our results are stronger.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

Feng-Rao bound for primary codes

{ b1, . . . , bn} a basis for Fn

q.

For c = t

i=1 ai

bi, at = 0 define ¯ ρ( c) = t.

• α ∗

β = (α1β1, . . . , αnβn). If ¯ ρ t

i=1 ai

bi

• ∗

bj

• = ¯

ρ( bi ∗ bj) = l for all choices of a1, . . . , at such that at = 0 then (i, j) is OWB. Theorem (The Feng-Rao bound for primary codes) If at = 0 then wH t

i=1 ai

bi

• is at least equal to the number of

l ∈ {1, . . . , n} for which a j exists with (i, j) OWB and ¯ ρ( bi ∗ bj) = l. Our results are stronger.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

The Feng-Rao bound for Dual codes

{ b1, . . . , bn} a basis for Fn

q.

Definition For c = 0, m( c) is the smallest index l such that c · bl = 0. Theorem (The Feng-Rao bound for dual codes) If m( c) = l then wH( c) is at least equal to the number i ∈ {1, . . . , n} for which a j exists with (i, j) OWB and ¯ ρ( bi ∗ bj) = l. Our results are stronger.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

The Feng-Rao bound for Dual codes

{ b1, . . . , bn} a basis for Fn

q.

Definition For c = 0, m( c) is the smallest index l such that c · bl = 0. Theorem (The Feng-Rao bound for dual codes) If m( c) = l then wH( c) is at least equal to the number i ∈ {1, . . . , n} for which a j exists with (i, j) OWB and ¯ ρ( bi ∗ bj) = l. Our results are stronger.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

The Feng-Rao bound for Dual codes

{ b1, . . . , bn} a basis for Fn

q.

Definition For c = 0, m( c) is the smallest index l such that c · bl = 0. Theorem (The Feng-Rao bound for dual codes) If m( c) = l then wH( c) is at least equal to the number i ∈ {1, . . . , n} for which a j exists with (i, j) OWB and ¯ ρ( bi ∗ bj) = l. Our results are stronger.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

A general lemma

Lemma Consider c ∈ Fn

• q. Let U ⊆ Fn

q be a subspace of dimension δ such

that for all non-zero words u ∈ U for some v it holds that ( u ∗ v) · c = 0. Then wH( c) ≥ δ. Task: The Feng-Rao method gives a particular space U. Determine a larger space U.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

Our approach

I = (X 4 + X 2 + X) − (Y 6 + Y 5 + Y 3) ⊆ F8[X, Y ]. Iq = I + X 8 − X, Y 8 − Y . ∆≺w (Iq) = {N1, . . . , N32}, with N1 ≺w · · · ≺w N32. 14 17 20 23 12 15 18 21 10 13 16 19 8 11 14 17 6 9 12 15 4 7 10 13 2 5 8 11 3 6 9 N21 N26 N30 N32 N17 N23 N28 N31 N13 N19 N25 N29 N9 N15 N22 N27 N6 N11 N18 N24 N4 N8 N14 N20 N2 N5 N10 N16 N1 N3 N7 N12 Consider c with m( c) = 21. That is, l = 21 is the smallest index such that c · bl = 0. Note that w(N21) = 14 = w(N22). Case 1: c · b22 = 0. Case 2: c · b22 = 0.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

Our approach

I = (X 4 + X 2 + X) − (Y 6 + Y 5 + Y 3) ⊆ F8[X, Y ]. Iq = I + X 8 − X, Y 8 − Y . ∆≺w (Iq) = {N1, . . . , N32}, with N1 ≺w · · · ≺w N32. 14 17 20 23 12 15 18 21 10 13 16 19 8 11 14 17 6 9 12 15 4 7 10 13 2 5 8 11 3 6 9 N21 N26 N30 N32 N17 N23 N28 N31 N13 N19 N25 N29 N9 N15 N22 N27 N6 N11 N18 N24 N4 N8 N14 N20 N2 N5 N10 N16 N1 N3 N7 N12 Consider c with m( c) = 21. That is, l = 21 is the smallest index such that c · bl = 0. Note that w(N21) = 14 = w(N22). Case 1: c · b22 = 0. Case 2: c · b22 = 0.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

m( c) = 21

14 17 20 23 12 15 18 21 10 13 16 19 8 11 14 17 6 9 12 15 4 7 10 13 2 5 8 11 3 6 9 Case 1: c · b22 = 0 U = Span{ b1, b2, b4, b6, b9, b13, b17, b21,

• b3,

b12, b5, b16, b7, b10}. For each u ∈ U\{ 0} there exists a bj such that ( u ∗ bj) · c = 0. wH( c) ≥ #U = 14.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

m( c) = 21 – cont.

14 17 20 23 12 15 18 21 10 13 16 19 8 11 14 17 6 9 12 15 4 7 10 13 2 5 8 11 3 6 9 Case 2: c · b22 = 0 U = Span{ b1, b2, b4, b6, b9, b13, b17, b21,

• b3,

b5, b7, b9, b11}. For each u ∈ U\{ 0} there exists a bj such that ( u ∗ bj) · c = 0. wH( c) ≥ #U = 13.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

m( c) = 21 – cont.

14 17 20 23 12 15 18 21 10 13 16 19 8 11 14 17 6 9 12 15 4 7 10 13 2 5 8 11 3 6 9 Case 1 gave wH( c) ≥ 14 and case 2 gave wH( c) ≥ 13. Hence, wH( c) ≥ 13. The Feng-Rao bound with WB or WWB gives 8. The Feng-Rao bound with OWB gives 8+2=10. The advisory bound (Salazar et al.) gi- ves 8+2+2=12.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

The bounds

For some varieties the primary and the dual bound produces similar parameters. For other varieties they do not. The advisory bound does not seem to have a counter part for primary codes. In the most general formulation our bounds involve three bases (not only one). One can show correspondence between Feng-Rao bounds for primary and dual codes in this setting. For our new bounds a general correspondence is unclear. Code constructions: We can use the { b1, . . . , bx} as generators/parity-checks. Or we can choose those basis elements that optimize the code parameters.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

Higher weights

z1: the Feng-Rao bound with WB; z2 the Feng-Rao bound with WWB; z3: the Feng-Rao bound with OWB; z4: the advisory bound; z5: our new method.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

Another variety

I = (X 9 + X 3 + X) − (Y 12 + Y 10 + Y 4) ⊆ F27[X, Y ]. Code length n = 243. Feng-Rao Feng-Rao Feng-Rao ”Advisory Our WB WWB OWB bound” bound d1(C(75)) 15 15 21 29 33 d2(C(75)) 16 16 24 34 38 d1(C(76)) 15 15 21 33 36 d2(C(76)) 16 16 24 38 39 d1(C(83)) 16 16 24 34 38 d2(C(83)) 17 17 27 39 41

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

A method for constructing many examples

Definition An (Fqt, Fq)-polynomial is a polynomial F(T) ∈ Fqt[T] such that F(γ) ∈ Fq holds for all γ ∈ Fqt. Theorem Consider the cyclotomic coset Ci modulo qt − 1. Then F(T) =

s∈Ci X s is an (Fqt, Fq)-polynomial.

Corollary Let F(T) be a polynomial as in the above theorem and different from the trace-polynomial. Then TrFqt /Fq(X) − F(Y ) has exactly q2t−1 zeros. It is obvious to compare our new affine variety codes with Norm-Trace codes that are one-point AG codes.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

Improved codes over F8 of length n = 32. Using the trace-polynomial (degree 4) in combination with an (F8, F2)-polynomial of degree 6 we get the ◦s. The ∗s correspond to the norm-trace codes (degrees 4 and 7).

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

Improved codes over F16 of length n = 128. Using the trace-polynomial (degree 8) in combination with an (F8, F2)-polynomial of degree 10 we get the ◦s. The ∗s correspond to the norm-trace codes (degrees 8 and 15).

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

Improved codes over F32 of length n = 512. Using the trace-polynomial (degree 16) in combination with an (F32, F2)-polynomial of degree 20 we get the ◦s. Using instead an (F32, F2)-polynomial of degree 26 we get the ∗s. The +s correspond to the norm-trace codes (degrees 16 and 31).

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

Improved codes over F64 of length n = 2048. Upper graph: the trace-polynomial (degree 32) in combination with an (F64, F2)-polynomial of degree 42. Lower graph: the norm-trace codes (degrees 32 and 63).

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

Primary codes versus dual codes

When the footprint ∆≺w (Iq) is a box, the bounds for primary and dual codes produce similar results. When the footprint is not a box, they often produce different results.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation

Footprint not a box

(X 20 + X 18 + X 10 + X 9 + X 5) − (Y 26 + Y 22 + Y 21 + Y 13 + Y 11) ∈ F32[X, Y ]. Code length n = 512. The ◦s correspond to improved primary codes, the ∗s correspond to improved dual codes.

Olav Geil, Stefano Martin Affine variety codes are better than their reputation