Best Known (21, 35, s)-Nets in Base 32
(21, 35, 264)-Net over F32 — Constructive and digital
Digital (21, 35, 264)-net over F32, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 1, 33)-net over F32, using
- s-reduction based on digital (0, 1, s)-net over F32 with arbitrarily large s, using
- digital (0, 2, 33)-net over F32, using
- digital (0, 2, 33)-net over F32 (see above)
- digital (0, 2, 33)-net over F32 (see above)
- 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, 4, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 7, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 14, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 1, 33)-net over F32, using
(21, 35, 586)-Net in Base 32 — Constructive
(21, 35, 586)-net in base 32, using
- net defined by OOA [i] based on OOA(3235, 586, S32, 14, 14), using
- OA 7-folding and stacking [i] based on OA(3235, 4102, S32, 14), using
- discarding factors based on OA(3235, 4104, S32, 14), using
- discarding parts of the base [i] based on linear OA(6429, 4104, F64, 14) (dual of [4104, 4075, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- linear OA(6427, 4096, F64, 14) (dual of [4096, 4069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(6421, 4096, F64, 11) (dual of [4096, 4075, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(642, 8, F64, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- discarding parts of the base [i] based on linear OA(6429, 4104, F64, 14) (dual of [4104, 4075, 15]-code), using
- discarding factors based on OA(3235, 4104, S32, 14), using
- OA 7-folding and stacking [i] based on OA(3235, 4102, S32, 14), using
(21, 35, 2070)-Net over F32 — Digital
Digital (21, 35, 2070)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3235, 2070, F32, 14) (dual of [2070, 2035, 15]-code), using
- 1035 step Varšamov–Edel lengthening with (ri) = (3, 4 times 0, 1, 19 times 0, 1, 61 times 0, 1, 161 times 0, 1, 315 times 0, 1, 469 times 0) [i] based on linear OA(3227, 1027, F32, 14) (dual of [1027, 1000, 15]-code), using
- construction XX applied to C1 = C([1022,11]), C2 = C([0,12]), C3 = C1 + C2 = C([0,11]), and C∩ = C1 ∩ C2 = C([1022,12]) [i] based on
- linear OA(3225, 1023, F32, 13) (dual of [1023, 998, 14]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,11}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(3225, 1023, F32, 13) (dual of [1023, 998, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(3227, 1023, F32, 14) (dual of [1023, 996, 15]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,12}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(3223, 1023, F32, 12) (dual of [1023, 1000, 13]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [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([1022,11]), C2 = C([0,12]), C3 = C1 + C2 = C([0,11]), and C∩ = C1 ∩ C2 = C([1022,12]) [i] based on
- 1035 step Varšamov–Edel lengthening with (ri) = (3, 4 times 0, 1, 19 times 0, 1, 61 times 0, 1, 161 times 0, 1, 315 times 0, 1, 469 times 0) [i] based on linear OA(3227, 1027, F32, 14) (dual of [1027, 1000, 15]-code), using
(21, 35, 3658528)-Net in Base 32 — Upper bound on s
There is no (21, 35, 3658529)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 47890 515753 012131 560122 202713 249913 060053 107004 918048 > 3235 [i]