Best Known (35−9, 35, s)-Nets in Base 32
(35−9, 35, 262147)-Net over F32 — Constructive and digital
Digital (26, 35, 262147)-net over F32, using
- net defined by OOA [i] based on linear OOA(3235, 262147, F32, 9, 9) (dual of [(262147, 9), 2359288, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(3235, 1048589, F32, 9) (dual of [1048589, 1048554, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(3235, 1048590, F32, 9) (dual of [1048590, 1048555, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(3233, 1048576, F32, 9) (dual of [1048576, 1048543, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(3221, 1048576, F32, 6) (dual of [1048576, 1048555, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(322, 14, F32, 2) (dual of [14, 12, 3]-code or 14-arc in PG(1,32)), using
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- Reed–Solomon code RS(30,32) [i]
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(3235, 1048590, F32, 9) (dual of [1048590, 1048555, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(3235, 1048589, F32, 9) (dual of [1048589, 1048554, 10]-code), using
(35−9, 35, 524288)-Net in Base 32 — Constructive
(26, 35, 524288)-net in base 32, using
- base change [i] based on digital (16, 25, 524288)-net over F128, using
- net defined by OOA [i] based on linear OOA(12825, 524288, F128, 9, 9) (dual of [(524288, 9), 4718567, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(12825, 2097153, F128, 9) (dual of [2097153, 2097128, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(12825, 2097153, F128, 9) (dual of [2097153, 2097128, 10]-code), using
- net defined by OOA [i] based on linear OOA(12825, 524288, F128, 9, 9) (dual of [(524288, 9), 4718567, 10]-NRT-code), using
(35−9, 35, 1048590)-Net over F32 — Digital
Digital (26, 35, 1048590)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3235, 1048590, F32, 9) (dual of [1048590, 1048555, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(3233, 1048576, F32, 9) (dual of [1048576, 1048543, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(3221, 1048576, F32, 6) (dual of [1048576, 1048555, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(322, 14, F32, 2) (dual of [14, 12, 3]-code or 14-arc in PG(1,32)), using
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- Reed–Solomon code RS(30,32) [i]
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
(35−9, 35, large)-Net in Base 32 — Upper bound on s
There is no (26, 35, large)-net in base 32, because
- 7 times m-reduction [i] would yield (26, 28, large)-net in base 32, but