Best Known (25, 25+18, s)-Nets in Base 32
(25, 25+18, 203)-Net over F32 — Constructive and digital
Digital (25, 43, 203)-net over F32, using
- 1 times m-reduction [i] based on digital (25, 44, 203)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 16, 99)-net over F32, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 3, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 0 and N(F) ≥ 33, using
- the rational function field F32(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- digital (0, 4, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 9, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 3, 33)-net over F32, using
- generalized (u, u+v)-construction [i] based on
- digital (9, 28, 104)-net over F32, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- digital (7, 16, 99)-net over F32, using
- (u, u+v)-construction [i] based on
(25, 25+18, 455)-Net in Base 32 — Constructive
(25, 43, 455)-net in base 32, using
- 321 times duplication [i] based on (24, 42, 455)-net in base 32, using
- base change [i] based on digital (17, 35, 455)-net over F64, using
- net defined by OOA [i] based on linear OOA(6435, 455, F64, 18, 18) (dual of [(455, 18), 8155, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(6435, 4095, F64, 18) (dual of [4095, 4060, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(6435, 4095, F64, 18) (dual of [4095, 4060, 19]-code), using
- net defined by OOA [i] based on linear OOA(6435, 455, F64, 18, 18) (dual of [(455, 18), 8155, 19]-NRT-code), using
- base change [i] based on digital (17, 35, 455)-net over F64, using
(25, 25+18, 1499)-Net over F32 — Digital
Digital (25, 43, 1499)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3243, 1499, F32, 18) (dual of [1499, 1456, 19]-code), using
- 464 step Varšamov–Edel lengthening with (ri) = (3, 1, 5 times 0, 1, 18 times 0, 1, 55 times 0, 1, 137 times 0, 1, 243 times 0) [i] based on linear OA(3235, 1027, F32, 18) (dual of [1027, 992, 19]-code), using
- construction XX applied to C1 = C([1022,15]), C2 = C([0,16]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([1022,16]) [i] based on
- linear OA(3233, 1023, F32, 17) (dual of [1023, 990, 18]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,15}, and designed minimum distance d ≥ |I|+1 = 18 [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(3235, 1023, F32, 18) (dual of [1023, 988, 19]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,16}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3231, 1023, F32, 16) (dual of [1023, 992, 17]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 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
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([1022,15]), C2 = C([0,16]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([1022,16]) [i] based on
- 464 step Varšamov–Edel lengthening with (ri) = (3, 1, 5 times 0, 1, 18 times 0, 1, 55 times 0, 1, 137 times 0, 1, 243 times 0) [i] based on linear OA(3235, 1027, F32, 18) (dual of [1027, 992, 19]-code), using
(25, 25+18, 2078073)-Net in Base 32 — Upper bound on s
There is no (25, 43, 2078074)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 52656 181193 713987 711175 556091 589820 583701 038228 697997 938533 423677 > 3243 [i]