The Feng Rao bounds. KIAS International Conference on Coding Theory and Applications Olav Geil, Aalborg University, Denmark

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1 Olav Geil Aalborg University Denmark KIAS International Conference on Coding Theory and Applications 2012

2 Linear code = a subspace. Operations are: Vector addition. Scalar multiplication. [n, k, d] the usual parameters. To deal with d (and k and even n) the compontwise product is useful: (c 1,..., c n ) (d 1,..., d n ) = (c 1 d 1,..., c n d n ).

3 Linear code = a subspace. Operations are: Vector addition. Scalar multiplication. [n, k, d] the usual parameters. To deal with d (and k and even n) the compontwise product is useful: (c 1,..., c n ) (d 1,..., d n ) = (c 1 d 1,..., c n d n ).

4 Claim: Code constructions with a supporting algebra: algebraic geometric codes, Reed Muller codes and relatives, affine variety codes, are about getting information on the componentwise product. Multiplication in higher algebra ( ) Multiplication in quotient ring Componentwise product at code level

5 Claim: Code constructions with a supporting algebra: algebraic geometric codes, Reed Muller codes and relatives, affine variety codes, are about getting information on the componentwise product. Multiplication in higher algebra ( ) Multiplication in quotient ring Componentwise product at code level

6 Dual code Parity check matrix Primary code Generator matrix The usual Feng Rao bound (Feng Rao bound for dual codes) Order bound The Andersen G bound (Feng Rao bound for primary codes) Footprint bound

7 Dual code Parity check matrix Primary code Generator matrix The usual Feng Rao bound (Feng Rao bound for dual codes) Order bound The Andersen G bound (Feng Rao bound for primary codes) Footprint bound

8 Dual code Parity check matrix Primary code Generator matrix The usual Feng Rao bound (Feng Rao bound for dual codes) Order bound The Andersen G bound (Feng Rao bound for primary codes) Footprint bound

9 This talk: connection between the levels of description, connection between dual and primary Results: Consequences of the above connections. Information derived from medium and low level descriptions. Important results that are not covered: Higher level results such as Beelen bound, Duursma Kirov Park bound and list decoding of algebraic geometric codes by Lee Bras-Amorós O Sullivan s method.

10 This talk: connection between the levels of description, connection between dual and primary Results: Consequences of the above connections. Information derived from medium and low level descriptions. Important results that are not covered: Higher level results such as Beelen bound, Duursma Kirov Park bound and list decoding of algebraic geometric codes by Lee Bras-Amorós O Sullivan s method.

11 This talk: connection between the levels of description, connection between dual and primary Results: Consequences of the above connections. Information derived from medium and low level descriptions. Important results that are not covered: Higher level results such as Beelen bound, Duursma Kirov Park bound and list decoding of algebraic geometric codes by Lee Bras-Amorós O Sullivan s method.

12 Ideal J F[ X ] The footprint: (J) = { X α X α is not a leading monomial of any polynomial in J} I F q [ X ], I q = I + X q 1 X 1,..., X q m X m. The footprint bound in a special case: #V Fq (I q ) = # (I q ).

13 Ideal J F[ X ] The footprint: (J) = { X α X α is not a leading monomial of any polynomial in J} I F q [ X ], I q = I + X q 1 X 1,..., X q m X m. The footprint bound in a special case: #V Fq (I q ) = # (I q ).

14 I q = X q X, Y q Y (I q ) = {X i Y j 0 i, j < q} #V Fq (I q ) = q 2 I q 2 = X q+1 Y q Y, X q2 X, Y q2 Y. Choose monomial ordering with x q+1 Y q, (I q 2) {X i Y j 0 i < q 2, 0 j < q} #V (I Fq 2 q 2) q q2 = q 3 Study of norm/trace gives q 3 zeros.

15 I q = X q X, Y q Y (I q ) = {X i Y j 0 i, j < q} #V Fq (I q ) = q 2 I q 2 = X q+1 Y q Y, X q2 X, Y q2 Y. Choose monomial ordering with x q+1 Y q, (I q 2) {X i Y j 0 i < q 2, 0 j < q} #V (I Fq 2 q 2) q q2 = q 3 Study of norm/trace gives q 3 zeros.

16 Gröbner basis for J w.r.t. is a basis for J such that (J) can easily be read off. G = {G 1,..., G s } J is Gröbner basis for J w.r.t. iff any monomial in lm(j) is divisible by some lm(g i ). Gröbner basis for I q gives exact information on #V Fq (I q ).

17 Gröbner basis for J w.r.t. is a basis for J such that (J) can easily be read off. G = {G 1,..., G s } J is Gröbner basis for J w.r.t. iff any monomial in lm(j) is divisible by some lm(g i ). Gröbner basis for I q gives exact information on #V Fq (I q ).

18 V Fq (I q ) = {P 1,..., P n }. Codeword c = (F (P 1 ),..., F (P n )). w H ( c) = n # (I q + F ) (n minus number of commen zeros). Information on which leading monomials occour in the code construction gives information on minimum distance. Improved code construction straight forward.

19 V Fq (I q ) = {P 1,..., P n }. Codeword c = (F (P 1 ),..., F (P n )). w H ( c) = n # (I q + F ) (n minus number of commen zeros). Information on which leading monomials occour in the code construction gives information on minimum distance. Improved code construction straight forward.

20 V Fq (I q ) = {P 1,..., P n }. Codeword c = (F (P 1 ),..., F (P n )). w H ( c) = n # (I q + F ) (n minus number of commen zeros). Information on which leading monomials occour in the code construction gives information on minimum distance. Improved code construction straight forward.

21 {M + J M (J)} is a basis for F[ X ]/J as a vectorspace over F. G = M 1 (P 1 ) M 1 (P n ) M 2 (P 1 ) M 2 (P n )..... M k (P 1 ) M k (P n ), M 1,..., M k (I q ), M i M j is a code of dimension k

22 {M + J M (J)} is a basis for F[ X ]/J as a vectorspace over F. G = M 1 (P 1 ) M 1 (P n ) M 2 (P 1 ) M 2 (P n )..... M k (P 1 ) M k (P n ), M 1,..., M k (I q ), M i M j is a code of dimension k

23 Reed Muller codes: Let I 5 = X 5 X, Y 5 Y and c = (F (P 1 ),..., F (P n=25 )), with lm(f ) = X i Y j. We get w H ( c) = n # (I 5 + F ) (5 i)(5 j). Y 4 XY 4 X 2 Y 4 X 3 Y 4 X 4 Y 4 Y 3 XY 3 X 2 Y 3 X 3 Y 3 X 4 Y 3 Y 2 XY 2 X 2 Y 2 X 3 Y 2 X 4 Y 2 Y XY X 2 Y X 3 Y X 4 Y 1 X X 2 X 3 X RM 5 (4, 2) is [25, 15, 5] Improved code construction gives [25, 17, 5]

24 Reed Muller codes: Let I 5 = X 5 X, Y 5 Y and c = (F (P 1 ),..., F (P n=25 )), with lm(f ) = X i Y j. We get w H ( c) = n # (I 5 + F ) (5 i)(5 j). Y 4 XY 4 X 2 Y 4 X 3 Y 4 X 4 Y 4 Y 3 XY 3 X 2 Y 3 X 3 Y 3 X 4 Y 3 Y 2 XY 2 X 2 Y 2 X 3 Y 2 X 4 Y 2 Y XY X 2 Y X 3 Y X 4 Y 1 X X 2 X 3 X RM 5 (4, 2) is [25, 15, 5] Improved code construction gives [25, 17, 5]

25 Reed Muller codes: Let I 5 = X 5 X, Y 5 Y and c = (F (P 1 ),..., F (P n=25 )), with lm(f ) = X i Y j. We get w H ( c) = n # (I 5 + F ) (5 i)(5 j). Y 4 XY 4 X 2 Y 4 X 3 Y 4 X 4 Y 4 Y 3 XY 3 X 2 Y 3 X 3 Y 3 X 4 Y 3 Y 2 XY 2 X 2 Y 2 X 3 Y 2 X 4 Y 2 Y XY X 2 Y X 3 Y X 4 Y 1 X X 2 X 3 X RM 5 (4, 2) is [25, 15, 5] Improved code construction gives [25, 17, 5]

26 Hermitian codes: I = X q+1 Y q Y, I q 2 = I + X q2 X, Y q2 Y. w(x i Y j ) = iq + j(q + 1) X s Y t w X u Y v if w(x s Y t ) < w(x u Y v ) or w(x s Y t ) = w(x u Y v ) and t < v Weighted degree lexicographic ordering.

27 I 4 = X 3 Y 2 Y, X 4 X, Y 4 Y. w (I 4 ) Y XY X 2 Y X 3 Y X X 2 X c = (F (P 1 ),..., F (P 8 )) lm(f ) = Y w H ( c) = #{M w (I 4 ) M / w (I 4 + F )}. YF rem X 3 Y 2 Y = Y (Y + ) rem X 3 Y 2 Y = X 3 + w H ( c) #w ( w (I 4 )) (w(y ) + w( w (I 4 ))). (what we hit is what we get).

28 I 4 = X 3 Y 2 Y, X 4 X, Y 4 Y. w (I 4 ) Y XY X 2 Y X 3 Y X X 2 X c = (F (P 1 ),..., F (P 8 )) lm(f ) = Y w H ( c) = #{M w (I 4 ) M / w (I 4 + F )}. YF rem X 3 Y 2 Y = Y (Y + ) rem X 3 Y 2 Y = X 3 + w H ( c) #w ( w (I 4 )) (w(y ) + w( w (I 4 ))). (what we hit is what we get).

29 I 9 = X 4 Y 3 Y, X 9 X, Y 9 Y. w(x ) = 3, w(y ) = 4. Y 2 XY 2 X 2 Y 2 x 3 Y 2 X 4 Y 2 X 5 Y 2 X 6 Y 2 X 7 Y 2 X 8 Y 2 Y XY X 2 Y x 3 Y X 4 Y X 5 Y X 6 Y X 7 Y X 8 Y 1 X X 2 x 3 X 4 X 5 X 6 X 7 X

30 One point algebraic geometric codes: P 1,..., P n, Q rational places of function field over F q. To construct C L (D = P P n, vq) we need basis for: v s=0 L(sQ) s=0 L(sQ). Everything, can be translated into affine variety description: s=0 L(sQ) = F q[x 1,..., X m ]/I {P 1,..., P n } V Fq (I ). Affine variety description includes determination of minimum distance via footprint bound.

31 One point algebraic geometric codes: P 1,..., P n, Q rational places of function field over F q. To construct C L (D = P P n, vq) we need basis for: v s=0 L(sQ) s=0 L(sQ). Everything, can be translated into affine variety description: s=0 L(sQ) = F q[x 1,..., X m ]/I {P 1,..., P n } V Fq (I ). Affine variety description includes determination of minimum distance via footprint bound.

32 Weierstrass semigroup: H(Q) = ν Q ( s=0 L(sQ) ) = w 1,..., w m. Definition: Given weights w 1,..., w m define w( X α ) = α 1 w α m w m. Define w by X α w X β if w( X α ) < w( X β ) or w( X α ) = w( X β ) but X α M X β ( M can be anything, for instance lex ) Example: w(x ) = q, w(y ) = q + 1, M = lex with X lex Y. F (X, Y ) = X q+1 Y q Y, w(x q+1 ) = w(y q ) = q(q + 1) and lm(f ) = Y q.

33 Weierstrass semigroup: H(Q) = ν Q ( s=0 L(sQ) ) = w 1,..., w m. Definition: Given weights w 1,..., w m define w( X α ) = α 1 w α m w m. Define w by X α w X β if w( X α ) < w( X β ) or w( X α ) = w( X β ) but X α M X β ( M can be anything, for instance lex ) Example: w(x ) = q, w(y ) = q + 1, M = lex with X lex Y. F (X, Y ) = X q+1 Y q Y, w(x q+1 ) = w(y q ) = q(q + 1) and lm(f ) = Y q.

34 Order domain conditions: I = F 1 ( X ),..., F s ( X ) F[ X ] and w 1,..., w m satisfy ODC if: 1. {F 1,..., F s } is a Gröbner basis w.r.t. w. 2. F i, i = 1,..., s contains exactly two monomials of highest weight. 3. No two monomials in w ( F 1,..., F s ) are of the same weight. Example: I = X q+1 Y q Y F q 2[X, Y ] 1. OK 2. OK 3. w (I ) = {X i Y j 0 j < q, 0 i} OK

35 Order domain conditions: I = F 1 ( X ),..., F s ( X ) F[ X ] and w 1,..., w m satisfy ODC if: 1. {F 1,..., F s } is a Gröbner basis w.r.t. w. 2. F i, i = 1,..., s contains exactly two monomials of highest weight. 3. No two monomials in w ( F 1,..., F s ) are of the same weight. Example: I = X q+1 Y q Y F q 2[X, Y ] 1. OK 2. OK 3. w (I ) = {X i Y j 0 j < q, 0 i} OK

36 Theorem (Miura 1997, Pellikaan 2001): s=0 L(sQ) = F[ X ]/I where I and corresponding weights satisfy order domain conditions. Corollary: C L (P P n, vq) = Span Fq {(M(P 1 ),..., M(P n )) M w (I q ), w(m) v}. Footprint method better than Goppa bound. (Andersen-G)

37 Theorem (Miura 1997, Pellikaan 2001): s=0 L(sQ) = F[ X ]/I where I and corresponding weights satisfy order domain conditions. Corollary: C L (P P n, vq) = Span Fq {(M(P 1 ),..., M(P n )) M w (I q ), w(m) v}. Footprint method better than Goppa bound. (Andersen-G)

38 Theorem (Miura 1997, Pellikaan 2001): s=0 L(sQ) = F[ X ]/I where I and corresponding weights satisfy order domain conditions. Corollary: C L (P P n, vq) = Span Fq {(M(P 1 ),..., M(P n )) M w (I q ), w(m) v}. Footprint method better than Goppa bound. (Andersen-G)

39 Weierstrass semigroup Λ = λ 1,..., λ m. Corollary: (G Matsumoto 2009) A function field having Λ as a Weierstrass semigroup can at most have ( ) # Λ\ m i=1 (qλ i + Λ) + 1 rational places. Term qλ i comes from X q i X i. Term +1 corresponds to the place with Weierstrass semigroup Λ. Better than Serre bound for small q. Gives a way for excluding possible Weierstrass semigroups when genus and number of zeros are known.

40 Order domains are generalizations of s=0 L(sQ). For transcendence degree r, weights are in N r 0 (when finitely generated) G Pellikaan Gives a way of generalizing algebraic geometric codes to higher transcendence degree. Think of Reed Muller code as higher transcendence degree version of Reed Solomon code. Order domain conditions and Pellikaan Miura correspondence also work for higher transcendence degrees G Pellikaan So does methods for estimating parameters. Descriptions can be abstract or be given as concrete quotient ring.

41 Order domains are generalizations of s=0 L(sQ). For transcendence degree r, weights are in N r 0 (when finitely generated) G Pellikaan Gives a way of generalizing algebraic geometric codes to higher transcendence degree. Think of Reed Muller code as higher transcendence degree version of Reed Solomon code. Order domain conditions and Pellikaan Miura correspondence also work for higher transcendence degrees G Pellikaan So does methods for estimating parameters. Descriptions can be abstract or be given as concrete quotient ring.

42 Order domains are generalizations of s=0 L(sQ). For transcendence degree r, weights are in N r 0 (when finitely generated) G Pellikaan Gives a way of generalizing algebraic geometric codes to higher transcendence degree. Think of Reed Muller code as higher transcendence degree version of Reed Solomon code. Order domain conditions and Pellikaan Miura correspondence also work for higher transcendence degrees G Pellikaan So does methods for estimating parameters. Descriptions can be abstract or be given as concrete quotient ring.

43 Order domains are generalizations of s=0 L(sQ). For transcendence degree r, weights are in N r 0 (when finitely generated) G Pellikaan Gives a way of generalizing algebraic geometric codes to higher transcendence degree. Think of Reed Muller code as higher transcendence degree version of Reed Solomon code. Order domain conditions and Pellikaan Miura correspondence also work for higher transcendence degrees G Pellikaan So does methods for estimating parameters. Descriptions can be abstract or be given as concrete quotient ring.

44 Order domains are generalizations of s=0 L(sQ). For transcendence degree r, weights are in N r 0 (when finitely generated) G Pellikaan Gives a way of generalizing algebraic geometric codes to higher transcendence degree. Think of Reed Muller code as higher transcendence degree version of Reed Solomon code. Order domain conditions and Pellikaan Miura correspondence also work for higher transcendence degrees G Pellikaan So does methods for estimating parameters. Descriptions can be abstract or be given as concrete quotient ring.

45 The footprint-method applied to order domain conditions: I = F 1 ( X,..., F s ( X ), w (I q ) = {M 1,..., M n }. c = ev(f ), lm(f ) = M i. w H ( c) = # ( w (I q )\ w (I q + F ) ) = #{M w (I q ) M is a leading monomial of a polynomial in I q + F } # monomials in w (I q ) hit by M i (using F 1,..., F s ) = # ( w( w (I q )) ( w(m i ) + w( w (I q )) )).

46 Linear code level: B = { b 1,..., b n } and U = { u 1,..., u n } bases for F n q. { 0} = L 0 L 1 = Span{ b 1 } L 2 = Span{ b 1, b 2 } L n = F n q. ρ B ( c) = i, if c L i \L i 1. (i, j) is OWB if ρ B ( b i u j ) < ρ B ( b i u j ) for i = 1,..., i 1. If a supporting algebra is given then information can be extracted regarding above. Think of B = U corresponding to {1, X, X 2,..., X q 1 }. ev(x i ) ev(x j ) = ev(x i+j ) applied when i + j < q.

47 Linear code level: B = { b 1,..., b n } and U = { u 1,..., u n } bases for F n q. { 0} = L 0 L 1 = Span{ b 1 } L 2 = Span{ b 1, b 2 } L n = F n q. ρ B ( c) = i, if c L i \L i 1. (i, j) is OWB if ρ B ( b i u j ) < ρ B ( b i u j ) for i = 1,..., i 1. If a supporting algebra is given then information can be extracted regarding above. Think of B = U corresponding to {1, X, X 2,..., X q 1 }. ev(x i ) ev(x j ) = ev(x i+j ) applied when i + j < q.

48 Linear code level: B = { b 1,..., b n } and U = { u 1,..., u n } bases for F n q. { 0} = L 0 L 1 = Span{ b 1 } L 2 = Span{ b 1, b 2 } L n = F n q. ρ B ( c) = i, if c L i \L i 1. (i, j) is OWB if ρ B ( b i u j ) < ρ B ( b i u j ) for i = 1,..., i 1. If a supporting algebra is given then information can be extracted regarding above. Think of B = U corresponding to {1, X, X 2,..., X q 1 }. ev(x i ) ev(x j ) = ev(x i+j ) applied when i + j < q.

49 To hit: σ(i) = #{l j such that (i, j) is OWB and ρ B ( b i u j ) = l} Theorem: If ρ B ( c) = i then w H ( c) σ(i). Proof: Assume (i, j 1 ), (i, j 2 ),..., (i, j σ ) hits l 1, l 2,..., l σ. { c u j1,, c u jσ } is linearly independent. Hence, c Span{ u j1, u jσ } is of dimension σ. But { c d d F n q} is of dimension w H ( c).

50 To hit: σ(i) = #{l j such that (i, j) is OWB and ρ B ( b i u j ) = l} Theorem: If ρ B ( c) = i then w H ( c) σ(i). Proof: Assume (i, j 1 ), (i, j 2 ),..., (i, j σ ) hits l 1, l 2,..., l σ. { c u j1,, c u jσ } is linearly independent. Hence, c Span{ u j1, u jσ } is of dimension σ. But { c d d F n q} is of dimension w H ( c).

51 To hit: σ(i) = #{l j such that (i, j) is OWB and ρ B ( b i u j ) = l} Theorem: If ρ B ( c) = i then w H ( c) σ(i). Proof: Assume (i, j 1 ), (i, j 2 ),..., (i, j σ ) hits l 1, l 2,..., l σ. { c u j1,, c u jσ } is linearly independent. Hence, c Span{ u j1, u jσ } is of dimension σ. But { c d d F n q} is of dimension w H ( c).

52 To be hit: µ(l) = #{i j such that (i, j) is OWB Theorem: and ρ B ( b i u j ) = l} Let l be such that c b l 0 but c b l = 0 for all l < l. Then w H ( c) µ(l). Proof: Same type of arguments as before. Primary code: minimum distance smallest σ(i) value among generating vectors. Dual code: minimum distance smallest µ value among non-parity-check vectors.

53 To be hit: µ(l) = #{i j such that (i, j) is OWB Theorem: and ρ B ( b i u j ) = l} Let l be such that c b l 0 but c b l = 0 for all l < l. Then w H ( c) µ(l). Proof: Same type of arguments as before. Primary code: minimum distance smallest σ(i) value among generating vectors. Dual code: minimum distance smallest µ value among non-parity-check vectors.

54 To be hit: µ(l) = #{i j such that (i, j) is OWB Theorem: and ρ B ( b i u j ) = l} Let l be such that c b l 0 but c b l = 0 for all l < l. Then w H ( c) µ(l). Proof: Same type of arguments as before. Primary code: minimum distance smallest σ(i) value among generating vectors. Dual code: minimum distance smallest µ value among non-parity-check vectors.

55 Recent results (with Matsumoto and Ruano): G = { g 1,..., g n }, H = { h 1,..., h n } and U = { u 1,..., u n } with g T ] 1. [ hn h 1 = I g T n then very nice translation between ρ G, σ with respect to (G, U) on the one side and ρ H, µ with respect to (H, U) on the other side. Primary code description dual code description. Feng Rao majority decoding algorithm for dual codes (usually described by means of algebra) can be formulated in linear code set-up (Matsumoto Miura 2000). Works for WB. Decoding of algebraically defined primary codes: Go to linear code level. Detect dual description and use linear version of decoding algorithm. Feng Rao bound for dual codes strongly related to footprint bound.

56 Recent results (with Matsumoto and Ruano): G = { g 1,..., g n }, H = { h 1,..., h n } and U = { u 1,..., u n } with g T ] 1. [ hn h 1 = I g T n then very nice translation between ρ G, σ with respect to (G, U) on the one side and ρ H, µ with respect to (H, U) on the other side. Primary code description dual code description. Feng Rao majority decoding algorithm for dual codes (usually described by means of algebra) can be formulated in linear code set-up (Matsumoto Miura 2000). Works for WB. Decoding of algebraically defined primary codes: Go to linear code level. Detect dual description and use linear version of decoding algorithm. Feng Rao bound for dual codes strongly related to footprint bound.

57 Recent results (with Matsumoto and Ruano): G = { g 1,..., g n }, H = { h 1,..., h n } and U = { u 1,..., u n } with g T ] 1. [ hn h 1 = I g T n then very nice translation between ρ G, σ with respect to (G, U) on the one side and ρ H, µ with respect to (H, U) on the other side. Primary code description dual code description. Feng Rao majority decoding algorithm for dual codes (usually described by means of algebra) can be formulated in linear code set-up (Matsumoto Miura 2000). Works for WB. Decoding of algebraically defined primary codes: Go to linear code level. Detect dual description and use linear version of decoding algorithm. Feng Rao bound for dual codes strongly related to footprint bound.

58 Recent results (with Matsumoto and Ruano): G = { g 1,..., g n }, H = { h 1,..., h n } and U = { u 1,..., u n } with g T ] 1. [ hn h 1 = I g T n then very nice translation between ρ G, σ with respect to (G, U) on the one side and ρ H, µ with respect to (H, U) on the other side. Primary code description dual code description. Feng Rao majority decoding algorithm for dual codes (usually described by means of algebra) can be formulated in linear code set-up (Matsumoto Miura 2000). Works for WB. Decoding of algebraically defined primary codes: Go to linear code level. Detect dual description and use linear version of decoding algorithm. Feng Rao bound for dual codes strongly related to footprint bound.

59 R = F 5 [X, Y ]. {P 1 = (1, 1), P 2 = (1, 2), P 3 = (1, 3), P 4 = (2, 1),..., P 9 = (3, 3)} F 2 5 g 1 = ev(1), g 2 = ev(x ), g 3 = ev(y ), g 4 = ev(x 2 ), g 5 = ev(xy ), g 6 = ev(y 2 ), g 7 = ev(x 2 Y ), g 8 = ev(xy 2 ), g 9 = ev(x 2 Y 2 ). h1 = ev(x 2 Y 2 + XY 2 + X 2 Y + XY ), h2 = ev(x 2 Y 2 + 3XY 2 + X 2 Y + Y 2 + 3XY + Y ), h3 = ev(x 2 Y 2 + XY 2 + 3X 2 Y + 3XY + X 2 + X ), h4 = ev(xy 2 + Y 2 + XY + Y ), h5 = ev(x 2 Y 2 + 3XY 2 + 3X 2 Y + Y 2 + 4XY + X 2 + 3Y + 3X + 1), h6 = ev(x 2 Y + XY + X 2 + X ), h7 = ev(xy 2 + Y 2 + 3XY + 3Y + X + 1), h8 = ev(x 2 Y + 3XY + X 2 + Y + 3X + 1), h9 = ev(xy + Y + X + 1).

60 R = F 5 [X, Y ]. {P 1 = (1, 1), P 2 = (1, 2), P 3 = (1, 3), P 4 = (2, 1),..., P 9 = (3, 3)} F 2 5 g 1 = ev(1), g 2 = ev(x ), g 3 = ev(y ), g 4 = ev(x 2 ), g 5 = ev(xy ), g 6 = ev(y 2 ), g 7 = ev(x 2 Y ), g 8 = ev(xy 2 ), g 9 = ev(x 2 Y 2 ). h1 = ev(x 2 Y 2 + XY 2 + X 2 Y + XY ), h2 = ev(x 2 Y 2 + 3XY 2 + X 2 Y + Y 2 + 3XY + Y ), h3 = ev(x 2 Y 2 + XY 2 + 3X 2 Y + 3XY + X 2 + X ), h4 = ev(xy 2 + Y 2 + XY + Y ), h5 = ev(x 2 Y 2 + 3XY 2 + 3X 2 Y + Y 2 + 4XY + X 2 + 3Y + 3X + 1), h6 = ev(x 2 Y + XY + X 2 + X ), h7 = ev(xy 2 + Y 2 + 3XY + 3Y + X + 1), h8 = ev(x 2 Y + 3XY + X 2 + Y + 3X + 1), h9 = ev(xy + Y + X + 1).

61 Information from function field theory, Gröbner basis theory, algebra, order domain theory translates easily to information on ρ and OWB, WWB or WB. Multiplication corresponds to componentwise product. Recent list decoding algorithms for algebraic geometric codes decode beoyond the bound for primary codes. (Lee Bras-Amorós O Sullivan 2011, G Matsumoto Ruano 2012, Lee Bras-Amorós O Sullivan 2012).

62 Information from function field theory, Gröbner basis theory, algebra, order domain theory translates easily to information on ρ and OWB, WWB or WB. Multiplication corresponds to componentwise product. Recent list decoding algorithms for algebraic geometric codes decode beoyond the bound for primary codes. (Lee Bras-Amorós O Sullivan 2011, G Matsumoto Ruano 2012, Lee Bras-Amorós O Sullivan 2012).

63 Everything said so far regarding minimum distance can be lifted to generalized Hamming weights. Supp(D) = {i {1,..., n} c i 0 for some c D}. d i (C) = min{#supp(d) D C, dim(d) = i} Give information about behaviour of Wiretap channel of type II. Secret sharing schemes.

64 Everything said so far regarding minimum distance can be lifted to generalized Hamming weights. Supp(D) = {i {1,..., n} c i 0 for some c D}. d i (C) = min{#supp(d) D C, dim(d) = i} Give information about behaviour of Wiretap channel of type II. Secret sharing schemes.

65 Generalized Reed Muller codes: Y 4 XY 4 X 2 Y 4 X 3 Y 4 X 4 Y 4 Y 3 XY 3 X 2 Y 3 X 3 Y 3 X 4 Y 3 Y 2 XY 2 X 2 Y 2 X 3 Y 2 X 4 Y 2 Y XY X 2 Y X 3 Y X 4 Y 1 X X 2 X 3 X Minimum distance corresponds to value on border (can be realized as product of linear factors). Second smallest weight: What happens if leading monomial is on the border, but minimal value is not realized? Use Buchberger s algorithm at a theoretical level. Second smallest weigth IS second smallest number above for degrees up to q m 1. G-2008 For m = 2 the degrees > q are easily solved. G-2008 Ericson-1974, Enough to know the case with two variables.

66 Generalized Reed Muller codes: Y 4 XY 4 X 2 Y 4 X 3 Y 4 X 4 Y 4 Y 3 XY 3 X 2 Y 3 X 3 Y 3 X 4 Y 3 Y 2 XY 2 X 2 Y 2 X 3 Y 2 X 4 Y 2 Y XY X 2 Y X 3 Y X 4 Y 1 X X 2 X 3 X Minimum distance corresponds to value on border (can be realized as product of linear factors). Second smallest weight: What happens if leading monomial is on the border, but minimal value is not realized? Use Buchberger s algorithm at a theoretical level. Second smallest weigth IS second smallest number above for degrees up to q m 1. G-2008 For m = 2 the degrees > q are easily solved. G-2008 Ericson-1974, Enough to know the case with two variables.

67 Conclusion: The variety of levels can sometimes help in realizing what is really going on. Lower level descriptions often captures what is going on, but might appear technical.