Best Known (55, 102, s)-Nets in Base 32
(55, 102, 260)-Net over F32 — Constructive and digital
Digital (55, 102, 260)-net over F32, using
- generalized (u, u+v)-construction [i] based on
- digital (3, 18, 64)-net over F32, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 3 and N(F) ≥ 64, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- digital (7, 30, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- digital (7, 54, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32 (see above)
- digital (3, 18, 64)-net over F32, using
(55, 102, 513)-Net in Base 32 — Constructive
(55, 102, 513)-net in base 32, using
- t-expansion [i] based on (46, 102, 513)-net in base 32, using
- 6 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
- base change [i] based on digital (28, 90, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- base change [i] based on digital (28, 90, 513)-net over F64, using
- 6 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
(55, 102, 1287)-Net over F32 — Digital
Digital (55, 102, 1287)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(32102, 1287, F32, 47) (dual of [1287, 1185, 48]-code), using
- 248 step Varšamov–Edel lengthening with (ri) = (5, 1, 0, 0, 1, 5 times 0, 1, 12 times 0, 1, 25 times 0, 1, 43 times 0, 1, 68 times 0, 1, 85 times 0) [i] based on linear OA(3290, 1027, F32, 47) (dual of [1027, 937, 48]-code), using
- construction XX applied to C1 = C([1022,44]), C2 = C([0,45]), C3 = C1 + C2 = C([0,44]), and C∩ = C1 ∩ C2 = C([1022,45]) [i] based on
- linear OA(3288, 1023, F32, 46) (dual of [1023, 935, 47]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,44}, and designed minimum distance d ≥ |I|+1 = 47 [i]
- linear OA(3288, 1023, F32, 46) (dual of [1023, 935, 47]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,45], and designed minimum distance d ≥ |I|+1 = 47 [i]
- linear OA(3290, 1023, F32, 47) (dual of [1023, 933, 48]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,45}, and designed minimum distance d ≥ |I|+1 = 48 [i]
- linear OA(3286, 1023, F32, 45) (dual of [1023, 937, 46]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,44], and designed minimum distance d ≥ |I|+1 = 46 [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([1022,44]), C2 = C([0,45]), C3 = C1 + C2 = C([0,44]), and C∩ = C1 ∩ C2 = C([1022,45]) [i] based on
- 248 step Varšamov–Edel lengthening with (ri) = (5, 1, 0, 0, 1, 5 times 0, 1, 12 times 0, 1, 25 times 0, 1, 43 times 0, 1, 68 times 0, 1, 85 times 0) [i] based on linear OA(3290, 1027, F32, 47) (dual of [1027, 937, 48]-code), using
(55, 102, 1237833)-Net in Base 32 — Upper bound on s
There is no (55, 102, 1237834)-net in base 32, because
- 1 times m-reduction [i] would yield (55, 101, 1237834)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 104 750149 691077 813943 654464 905036 639724 414198 904941 419299 714385 047364 756952 512187 234991 731642 953053 613422 650447 129562 646242 101635 941447 020985 892349 564304 > 32101 [i]