Best Known (62−27, 62, s)-Nets in Base 32
(62−27, 62, 240)-Net over F32 — Constructive and digital
Digital (35, 62, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 24, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 11 and N(F) ≥ 120, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- digital (11, 38, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 24, 120)-net over F32, using
(62−27, 62, 354)-Net in Base 32 — Constructive
(35, 62, 354)-net in base 32, using
- (u, u+v)-construction [i] based on
- (8, 21, 257)-net in base 32, using
- base change [i] based on (2, 15, 257)-net in base 128, using
- 1 times m-reduction [i] based on (2, 16, 257)-net in base 128, using
- base change [i] based on digital (0, 14, 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, 14, 257)-net over F256, using
- 1 times m-reduction [i] based on (2, 16, 257)-net in base 128, using
- base change [i] based on (2, 15, 257)-net in base 128, using
- (14, 41, 177)-net in base 32, using
- 1 times m-reduction [i] based on (14, 42, 177)-net in base 32, using
- base change [i] based on digital (7, 35, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- base change [i] based on digital (7, 35, 177)-net over F64, using
- 1 times m-reduction [i] based on (14, 42, 177)-net in base 32, using
- (8, 21, 257)-net in base 32, using
(62−27, 62, 1340)-Net over F32 — Digital
Digital (35, 62, 1340)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3262, 1340, F32, 27) (dual of [1340, 1278, 28]-code), using
- 304 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 7 times 0, 1, 18 times 0, 1, 44 times 0, 1, 87 times 0, 1, 140 times 0) [i] based on linear OA(3253, 1027, F32, 27) (dual of [1027, 974, 28]-code), using
- construction XX applied to C1 = C([1022,24]), C2 = C([0,25]), C3 = C1 + C2 = C([0,24]), and C∩ = C1 ∩ C2 = C([1022,25]) [i] based on
- linear OA(3251, 1023, F32, 26) (dual of [1023, 972, 27]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,24}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(3251, 1023, F32, 26) (dual of [1023, 972, 27]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,25], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(3253, 1023, F32, 27) (dual of [1023, 970, 28]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,25}, and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3249, 1023, F32, 25) (dual of [1023, 974, 26]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,24], and designed minimum distance d ≥ |I|+1 = 26 [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([1022,24]), C2 = C([0,25]), C3 = C1 + C2 = C([0,24]), and C∩ = C1 ∩ C2 = C([1022,25]) [i] based on
- 304 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 7 times 0, 1, 18 times 0, 1, 44 times 0, 1, 87 times 0, 1, 140 times 0) [i] based on linear OA(3253, 1027, F32, 27) (dual of [1027, 974, 28]-code), using
(62−27, 62, 2111883)-Net in Base 32 — Upper bound on s
There is no (35, 62, 2111884)-net in base 32, because
- 1 times m-reduction [i] would yield (35, 61, 2111884)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 65 185296 234452 288657 450859 312688 883820 242114 639410 390679 303896 679546 901656 503396 649457 317024 > 3261 [i]