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Good covering codes from algebraic curves

Irene Platoni - University of Trento (Italy)

Workshop on Algebraic Curves and Function Fields over aFinite Field

Perugia, 2-7 February 2015

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

Covering codes

C is an [n, k , d ]q-linear code t = ⌊ d−12 ⌋ → t-error correcting

Covering Radius of C

R(C ) := maxv∈Fn

d(v ,C ) ≥ t

R(C ) = t ⇒ C is perfect.

R(C ) = t + 1⇒ C is quasi-perfect.

Covering Density of C

µ(C ) := #C · size of a sphere of radius R(C )

qn≥ 1

Covering codes

R(C ) := maxv∈Fn

d(v ,C ) ≥ t

qn≥ 1

Covering codes

R(C ) := maxv∈Fn

d(v ,C ) ≥ t

qn≥ 1

Covering Codes with small density

r = n − k → codimension

µ(C ) =1 + n(q − 1) +

)(q − 1)2 + . . .+

)(q − 1)R(C)

Remark

The shorter is the code with fixed codimension r , minimum distance dand covering radius R(C ), the better are its covering properties.

ℓ(r , ρ, q)d := min n for which there exists C ⊂ Fnq with

R(C ) = ρ, n− k = r , d(C ) = d

Our approach −→ Use of Galois Geometry

µ(C ) =1 + n(q − 1) +

)(q − 1)2 + . . .+

)(q − 1)R(C)

Remark

R(C ) = ρ, n− k = r , d(C ) = d

µ(C ) =1 + n(q − 1) +

)(q − 1)2 + . . .+

)(q − 1)R(C)

Remark

R(C ) = ρ, n− k = r , d(C ) = d

Complete caps

Σ = Σ(N , q)

Galois space of dimension N over the finite field Fq

The set S ⊂ Σ is a

cap if it does not contain three collinear pointscomplete cap if every point in Σ \ S is collinear with two points in S

Complete caps

Σ = Σ(N , q)

Complete caps

Σ = Σ(N , q)

Complete caps

Σ = Σ(N , q)

Complete caps

Σ = Σ(N , q)

Complete caps

Σ = Σ(N , q)

Complete caps

Σ = Σ(N , q)

Complete arcs

Σ = Σ(N , q)

The set A ⊂ Σ is

an arc if no N + 1 points are in the same hyperplanecomplete arc if every point in Σ \ A is contained in a hyperplanegenerated by N points of A

When N = 2, plane caps coincide with plane arcs

Complete arcs

Σ = Σ(N , q)

The set A ⊂ Σ is

Complete arcs

Σ = Σ(N , q)

The set A ⊂ Σ is

Complete arcs

Σ = Σ(N , q)

The set A ⊂ Σ is

Complete arcs

Σ = Σ(N , q)

The set A ⊂ Σ is

Complete arcs

Σ = Σ(N , q)

The set A ⊂ Σ is

Classical complete arcs and caps

If q is odd, then:

an irreducible conic of PG(2, q) is a complete cap of size q + 1;

if q is even, then:

an irreducible conic of PG(2, q) plus its nucleus is a complete cap ofsize q + 2.

An elliptic quadric of PG(3, q) is a complete cap of size q2 + 1.

If 2 ≤ N ≤ q − 2, the normal rational curve of PG(N , q)

{(1 : t : . . . : tN

)| t ∈ Fq} ∪ {(0 : . . . : 0 : 1)}

is a complete arc of size q + 1.

If q is odd, then:

if q is even, then:

{(1 : t : . . . : tN

)| t ∈ Fq} ∪ {(0 : . . . : 0 : 1)}

If q is odd, then:

if q is even, then:

{(1 : t : . . . : tN

)| t ∈ Fq} ∪ {(0 : . . . : 0 : 1)}

If q is odd, then:

if q is even, then:

{(1 : t : . . . : tN

)| t ∈ Fq} ∪ {(0 : . . . : 0 : 1)}

Linear Codes and Projective Sets

Parity-checkmatrix of an

[n, n − r , d ]q-code←→ Proper n-sets of

points in PG(r − 1, q)

ρ = 2, d = 4 (quasi-perfect codes) ↔ complete n-caps in PG(r − 1, q).

ρ = r − 1, d = r + 1 (MDS codes) ↔ complete n-arcs in PG(r − 1, q)

Aim: Determine upper-bounds on ℓ(r , 2, q)4 and ℓ(r , r − 1, q)r+1

(based on some constructions of small complete arcs and caps fromplane cubic curves defined over Fq)

Classification of cubic curves with at least one

rational inflection (p > 3)

Y = X 3 XY = (X − 1)3

Y (X 2 − β) = 1 Y 2 = X 3 + AX + BIrene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

Classification of cubic curves with at least one

rational inflection (p > 3)

Y = X 3 XY = (X − 1)3

Y (X 2 − β) = 1 Y 2 = X 3 + AX + BIrene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

Outline

Complete caps from singular cubics

Complete caps in AG(3, q)

Complete arcs from elliptic curves

Plane caps from irreducible cubics

q = ph, with p > 3 a prime

X irreducible plane cubic

G := {non-singular Fq-rational points of X}(G ,⊕) abelian group with neutral elementan inflection point O

P ⊕ Q ⊕ T = O ⇔ P ,Q,T are collinear distinct points of G

Proposition

K 6 G, of index m

(3,m) = 1

Q ∈ G \ K⇒ S = K ⊕ Q is a plane cap

• Q• P

•P ⊕ Q

Proposition

K 6 G, of index m

(3,m) = 1

• Q• P

•P ⊕ Q

Proposition

K 6 G, of index m

(3,m) = 1

• Q• P

•P ⊕ Q

Proposition

K 6 G, of index m

(3,m) = 1

Algebraic Method

C1) Write S in an algebraically parametrized form

S = {(f (t), g(t))︸︷︷︸

| t ∈ B} ⊆ AG(2, q)

where B can be Fq or F∗q

C2) ∀ P = (a, b) off XC2.1) Construct a curve CP : FP(X ,Y ) = 0, where

FP(X ,Y ) = det

a b 1f (X ) g(X ) 1f (Y ) g(Y ) 1

is defined over Fq

Remark

P = (a, b) is collinear with two points Px ,Py in S if the algebraic curveCP : FP(X ,Y ) = 0 has a suitable Fq-rational point (x , y).

Algebraic Method

C2.2) Apply Hasse-Weil Theorem to the curve CP

Theorem (Hasse-Weil, 1948)

Let C be an irreducible curve of genus g , defined over Fq . Then

|C(Fq)| ≥ q + 1− 2g√q

C3) Add some points on X \ S to S

Requirements

CP is irreducible (or has an irreducible component defined over Fq)

The genus of CP is not too large with respect to q

Algebraic Method

C2.2) Apply Hasse-Weil Theorem to the curve CP

Theorem (Hasse-Weil, 1948)

Let C be an irreducible curve of genus g , defined over Fq . Then

|C(Fq)| ≥ q + 1− 2g√q

C3) Add some points on X \ S to S

Requirements

CP is irreducible (or has an irreducible component defined over Fq)

The genus of CP is not too large with respect to q

Cuspidal Case: Y = X 3

G is isomorphic to (Fq,+)

G −→ Fq, Qv =(v , v3

)7−→ v

The subgroup of index m (m a divisor of q)

K = {(L(t), L(t)3) | t ∈ Fq}

L(T ) =∏

α∈M

(T − α), M < (Fq,+), #M = m

A cosetS = {(L(t) + t, (L(t) + t)3)

︸︷︷︸

| t ∈ Fq}

The curve CP : FP(X ,Y ) = 0

FP(X ,Y ) = b + (L(X ) + t)(L(Y ) + t)2 +

(L(X ) + t)2(L(Y ) + t)− a((L(X ) + t)2

+(L(X ) + t)(L(Y ) + t) + (L(Y ) + t)2)

CP : FP(X ,Y ) = 0 is an absolutely irreducible curve, defined over Fq

Upper bound for the genus: 3m2 − 3m+ 1

Hasse-Weil bound