Best Known (64−28, 64, s)-Nets in Base 32
(64−28, 64, 240)-Net over F32 — Constructive and digital
Digital (36, 64, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 25, 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, 39, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 25, 120)-net over F32, using
(64−28, 64, 337)-Net in Base 32 — Constructive
(36, 64, 337)-net in base 32, using
- (u, u+v)-construction [i] based on
- (4, 18, 80)-net in base 32, using
- base change [i] based on digital (1, 15, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- base change [i] based on digital (1, 15, 80)-net over F64, using
- (18, 46, 257)-net in base 32, using
- 2 times m-reduction [i] based on (18, 48, 257)-net in base 32, using
- base change [i] based on digital (0, 30, 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, 30, 257)-net over F256, using
- 2 times m-reduction [i] based on (18, 48, 257)-net in base 32, using
- (4, 18, 80)-net in base 32, using
(64−28, 64, 1322)-Net over F32 — Digital
Digital (36, 64, 1322)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3264, 1322, F32, 28) (dual of [1322, 1258, 29]-code), using
- 286 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 6 times 0, 1, 17 times 0, 1, 41 times 0, 1, 82 times 0, 1, 132 times 0) [i] based on linear OA(3255, 1027, F32, 28) (dual of [1027, 972, 29]-code), using
- construction XX applied to C1 = C([1022,25]), C2 = C([0,26]), C3 = C1 + C2 = C([0,25]), and C∩ = C1 ∩ C2 = C([1022,26]) [i] based on
- 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(3253, 1023, F32, 27) (dual of [1023, 970, 28]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,26], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3255, 1023, F32, 28) (dual of [1023, 968, 29]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,26}, and designed minimum distance d ≥ |I|+1 = 29 [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(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([1022,25]), C2 = C([0,26]), C3 = C1 + C2 = C([0,25]), and C∩ = C1 ∩ C2 = C([1022,26]) [i] based on
- 286 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 6 times 0, 1, 17 times 0, 1, 41 times 0, 1, 82 times 0, 1, 132 times 0) [i] based on linear OA(3255, 1027, F32, 28) (dual of [1027, 972, 29]-code), using
(64−28, 64, 1481765)-Net in Base 32 — Upper bound on s
There is no (36, 64, 1481766)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 2 135987 180439 045602 526835 502377 837259 597297 763595 412218 968215 754759 823528 393201 649275 742517 080784 > 3264 [i]