Best Known (35−15, 35, s)-Nets in Base 32
(35−15, 35, 198)-Net over F32 — Constructive and digital
Digital (20, 35, 198)-net over F32, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 33)-net over F32, using
- 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, 3, 33)-net over F32 (see above)
- digital (0, 5, 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, 15, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
(35−15, 35, 585)-Net in Base 32 — Constructive
(20, 35, 585)-net in base 32, using
- net defined by OOA [i] based on OOA(3235, 585, S32, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(3235, 4096, S32, 15), using
- discarding factors based on OA(3235, 4098, S32, 15), using
- discarding parts of the base [i] based on linear OA(6429, 4098, F64, 15) (dual of [4098, 4069, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(6429, 4096, F64, 15) (dual of [4096, 4067, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 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(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- discarding parts of the base [i] based on linear OA(6429, 4098, F64, 15) (dual of [4098, 4069, 16]-code), using
- discarding factors based on OA(3235, 4098, S32, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(3235, 4096, S32, 15), using
(35−15, 35, 1217)-Net over F32 — Digital
Digital (20, 35, 1217)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3235, 1217, F32, 15) (dual of [1217, 1182, 16]-code), using
- 184 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 13 times 0, 1, 44 times 0, 1, 120 times 0) [i] based on linear OA(3229, 1027, F32, 15) (dual of [1027, 998, 16]-code), using
- construction XX applied to C1 = C([1022,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([1022,13]) [i] based on
- 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(3227, 1023, F32, 14) (dual of [1023, 996, 15]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(3229, 1023, F32, 15) (dual of [1023, 994, 16]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,13}, and designed minimum distance d ≥ |I|+1 = 16 [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(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([1022,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([1022,13]) [i] based on
- 184 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 13 times 0, 1, 44 times 0, 1, 120 times 0) [i] based on linear OA(3229, 1027, F32, 15) (dual of [1027, 998, 16]-code), using
(35−15, 35, 2229896)-Net in Base 32 — Upper bound on s
There is no (20, 35, 2229897)-net in base 32, because
- 1 times m-reduction [i] would yield (20, 34, 2229897)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 1496 577945 483769 046577 896712 506880 344153 930885 677048 > 3234 [i]