Best Known (27, 27+19, s)-Nets in Base 32
(27, 27+19, 231)-Net over F32 — Constructive and digital
Digital (27, 46, 231)-net over F32, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 33)-net over F32, using
- 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 (0, 19, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
(27, 27+19, 514)-Net in Base 32 — Constructive
(27, 46, 514)-net in base 32, using
- (u, u+v)-construction [i] based on
- (6, 15, 257)-net in base 32, using
- 1 times m-reduction [i] based on (6, 16, 257)-net in base 32, using
- base change [i] based on digital (0, 10, 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, 10, 257)-net over F256, using
- 1 times m-reduction [i] based on (6, 16, 257)-net in base 32, using
- (12, 31, 257)-net in base 32, using
- 1 times m-reduction [i] based on (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 (see above)
- base change [i] based on digital (0, 20, 257)-net over F256, using
- 1 times m-reduction [i] based on (12, 32, 257)-net in base 32, using
- (6, 15, 257)-net in base 32, using
(27, 27+19, 1721)-Net over F32 — Digital
Digital (27, 46, 1721)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3246, 1721, F32, 19) (dual of [1721, 1675, 20]-code), using
- 685 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, 1, 292 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
- 685 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, 1, 292 times 0) [i] based on linear OA(3237, 1027, F32, 19) (dual of [1027, 990, 20]-code), using
(27, 27+19, 4488892)-Net in Base 32 — Upper bound on s
There is no (27, 46, 4488893)-net in base 32, because
- 1 times m-reduction [i] would yield (27, 45, 4488893)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 53 919918 627829 167283 294711 939684 825680 130515 488035 524073 347272 686940 > 3245 [i]