Best Known (90−31, 90, s)-Nets in Base 32
(90−31, 90, 392)-Net over F32 — Constructive and digital
Digital (59, 90, 392)-net over F32, using
- generalized (u, u+v)-construction [i] based on
- digital (6, 13, 98)-net over F32, using
- s-reduction based on digital (6, 13, 342)-net over F32, using
- net defined by OOA [i] based on linear OOA(3213, 342, F32, 7, 7) (dual of [(342, 7), 2381, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(3213, 1027, F32, 7) (dual of [1027, 1014, 8]-code), using
- construction XX applied to C1 = C([1022,4]), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([1022,5]) [i] based on
- linear OA(3211, 1023, F32, 6) (dual of [1023, 1012, 7]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,4}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(3211, 1023, F32, 6) (dual of [1023, 1012, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(3213, 1023, F32, 7) (dual of [1023, 1010, 8]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,5}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(329, 1023, F32, 5) (dual of [1023, 1014, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [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([1022,4]), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([1022,5]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(3213, 1027, F32, 7) (dual of [1027, 1014, 8]-code), using
- net defined by OOA [i] based on linear OOA(3213, 342, F32, 7, 7) (dual of [(342, 7), 2381, 8]-NRT-code), using
- s-reduction based on digital (6, 13, 342)-net over F32, using
- digital (7, 17, 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, 22, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32 (see above)
- digital (7, 38, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32 (see above)
- digital (6, 13, 98)-net over F32, using
(90−31, 90, 1093)-Net in Base 32 — Constructive
(59, 90, 1093)-net in base 32, using
- net defined by OOA [i] based on OOA(3290, 1093, S32, 31, 31), using
- OOA 15-folding and stacking with additional row [i] based on OA(3290, 16396, S32, 31), using
- discarding parts of the base [i] based on linear OA(12864, 16396, F128, 31) (dual of [16396, 16332, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- linear OA(12861, 16385, F128, 31) (dual of [16385, 16324, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(12853, 16385, F128, 27) (dual of [16385, 16332, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(1283, 11, F128, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,128) or 11-cap in PG(2,128)), using
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- Reed–Solomon code RS(125,128) [i]
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- discarding parts of the base [i] based on linear OA(12864, 16396, F128, 31) (dual of [16396, 16332, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on OA(3290, 16396, S32, 31), using
(90−31, 90, 12746)-Net over F32 — Digital
Digital (59, 90, 12746)-net over F32, using
(90−31, 90, large)-Net in Base 32 — Upper bound on s
There is no (59, 90, large)-net in base 32, because
- 29 times m-reduction [i] would yield (59, 61, large)-net in base 32, but