Best Known (26, 26+19, s)-Nets in Base 32
(26, 26+19, 209)-Net over F32 — Constructive and digital
Digital (26, 45, 209)-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, 3, 33)-net over F32 (see above)
- digital (0, 4, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 6, 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 (1, 20, 44)-net over F32, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- digital (0, 3, 33)-net over F32, using
(26, 26+19, 455)-Net in Base 32 — Constructive
(26, 45, 455)-net in base 32, using
- net defined by OOA [i] based on OOA(3245, 455, S32, 19, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(3245, 4096, S32, 19), using
- discarding factors based on OA(3245, 4098, S32, 19), using
- discarding parts of the base [i] based on linear OA(6437, 4098, F64, 19) (dual of [4098, 4061, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(6437, 4096, F64, 19) (dual of [4096, 4059, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- discarding parts of the base [i] based on linear OA(6437, 4098, F64, 19) (dual of [4098, 4061, 20]-code), using
- discarding factors based on OA(3245, 4098, S32, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(3245, 4096, S32, 19), using
(26, 26+19, 1427)-Net over F32 — Digital
Digital (26, 45, 1427)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3245, 1427, F32, 19) (dual of [1427, 1382, 20]-code), using
- 392 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 0, 1, 14 times 0, 1, 44 times 0, 1, 114 times 0, 1, 211 times 0) [i] based on linear OA(3237, 1027, F32, 19) (dual of [1027, 990, 20]-code), using
- construction XX applied to C1 = C([1022,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([1022,17]) [i] based on
- 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(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(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 = {−1,0,…,17}, and designed minimum distance d ≥ |I|+1 = 20 [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(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([1022,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([1022,17]) [i] based on
- 392 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 0, 1, 14 times 0, 1, 44 times 0, 1, 114 times 0, 1, 211 times 0) [i] based on linear OA(3237, 1027, F32, 19) (dual of [1027, 990, 20]-code), using
(26, 26+19, 3054218)-Net in Base 32 — Upper bound on s
There is no (26, 45, 3054219)-net in base 32, because
- 1 times m-reduction [i] would yield (26, 44, 3054219)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 1 684997 546898 162980 918956 828242 112009 598249 785605 216235 500487 219190 > 3244 [i]