Best Known (25, 25+20, s)-Nets in Base 32
(25, 25+20, 196)-Net over F32 — Constructive and digital
Digital (25, 45, 196)-net over F32, using
- 2 times m-reduction [i] based on digital (25, 47, 196)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 18, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- digital (7, 29, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32 (see above)
- digital (7, 18, 98)-net over F32, using
- (u, u+v)-construction [i] based on
(25, 25+20, 322)-Net in Base 32 — Constructive
(25, 45, 322)-net in base 32, using
- 321 times duplication [i] based on (24, 44, 322)-net in base 32, using
- (u, u+v)-construction [i] based on
- (2, 12, 65)-net in base 32, using
- base change [i] based on digital (0, 10, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- base change [i] based on digital (0, 10, 65)-net over F64, using
- (12, 32, 257)-net in base 32, using
- base change [i] based on digital (0, 20, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base change [i] based on digital (0, 20, 257)-net over F256, using
- (2, 12, 65)-net in base 32, using
- (u, u+v)-construction [i] based on
(25, 25+20, 1087)-Net over F32 — Digital
Digital (25, 45, 1087)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3245, 1087, F32, 20) (dual of [1087, 1042, 21]-code), using
- 52 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 11 times 0, 1, 36 times 0) [i] based on linear OA(3240, 1030, F32, 20) (dual of [1030, 990, 21]-code), using
- construction XX applied to C1 = C([1021,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([1021,17]) [i] based on
- linear OA(3237, 1023, F32, 19) (dual of [1023, 986, 20]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−2,−1,…,16}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3235, 1023, F32, 18) (dual of [1023, 988, 19]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3239, 1023, F32, 20) (dual of [1023, 984, 21]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−2,−1,…,17}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(3233, 1023, F32, 17) (dual of [1023, 990, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(321, 5, F32, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, 32, F32, 1) (dual of [32, 31, 2]-code), using
- Reed–Solomon code RS(31,32) [i]
- discarding factors / shortening the dual code based on linear OA(321, 32, F32, 1) (dual of [32, 31, 2]-code), using
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction XX applied to C1 = C([1021,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([1021,17]) [i] based on
- 52 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 11 times 0, 1, 36 times 0) [i] based on linear OA(3240, 1030, F32, 20) (dual of [1030, 990, 21]-code), using
(25, 25+20, 866536)-Net in Base 32 — Upper bound on s
There is no (25, 45, 866537)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 53 919913 635213 879013 137631 798362 725624 379334 127018 527348 846172 705808 > 3245 [i]