Best Known (64−29, 64, s)-Nets in Base 32
(64−29, 64, 224)-Net over F32 — Constructive and digital
Digital (35, 64, 224)-net over F32, using
- 1 times m-reduction [i] based on digital (35, 65, 224)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (9, 24, 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 (11, 41, 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 (9, 24, 104)-net over F32, using
- (u, u+v)-construction [i] based on
(64−29, 64, 322)-Net in Base 32 — Constructive
(35, 64, 322)-net in base 32, using
- (u, u+v)-construction [i] based on
- (3, 17, 65)-net in base 32, using
- 1 times m-reduction [i] based on (3, 18, 65)-net in base 32, using
- base change [i] based on digital (0, 15, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- base change [i] based on digital (0, 15, 65)-net over F64, using
- 1 times m-reduction [i] based on (3, 18, 65)-net in base 32, using
- (18, 47, 257)-net in base 32, using
- 1 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
- 1 times m-reduction [i] based on (18, 48, 257)-net in base 32, using
- (3, 17, 65)-net in base 32, using
(64−29, 64, 1100)-Net over F32 — Digital
Digital (35, 64, 1100)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3264, 1100, F32, 29) (dual of [1100, 1036, 30]-code), using
- 66 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 6 times 0, 1, 16 times 0, 1, 38 times 0) [i] based on linear OA(3257, 1027, F32, 29) (dual of [1027, 970, 30]-code), using
- construction XX applied to C1 = C([1022,26]), C2 = C([0,27]), C3 = C1 + C2 = C([0,26]), and C∩ = C1 ∩ C2 = C([1022,27]) [i] based on
- 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(3255, 1023, F32, 28) (dual of [1023, 968, 29]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,27], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3257, 1023, F32, 29) (dual of [1023, 966, 30]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,27}, and designed minimum distance d ≥ |I|+1 = 30 [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(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([1022,26]), C2 = C([0,27]), C3 = C1 + C2 = C([0,26]), and C∩ = C1 ∩ C2 = C([1022,27]) [i] based on
- 66 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 6 times 0, 1, 16 times 0, 1, 38 times 0) [i] based on linear OA(3257, 1027, F32, 29) (dual of [1027, 970, 30]-code), using
(64−29, 64, 1156826)-Net in Base 32 — Upper bound on s
There is no (35, 64, 1156827)-net in base 32, because
- 1 times m-reduction [i] would yield (35, 63, 1156827)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 66749 762900 438792 768341 943574 298742 864422 586001 629355 823969 494392 880207 902723 864219 235020 012532 > 3263 [i]