Best Known (23, 31, s)-Nets in Base 32
(23, 31, 262147)-Net over F32 — Constructive and digital
Digital (23, 31, 262147)-net over F32, using
- net defined by OOA [i] based on linear OOA(3231, 262147, F32, 8, 8) (dual of [(262147, 8), 2097145, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(3231, 1048588, F32, 8) (dual of [1048588, 1048557, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(3231, 1048590, F32, 8) (dual of [1048590, 1048559, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- linear OA(3229, 1048576, F32, 8) (dual of [1048576, 1048547, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(3217, 1048576, F32, 5) (dual of [1048576, 1048559, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(7) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(3231, 1048590, F32, 8) (dual of [1048590, 1048559, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(3231, 1048588, F32, 8) (dual of [1048588, 1048557, 9]-code), using
(23, 31, 524288)-Net in Base 32 — Constructive
(23, 31, 524288)-net in base 32, using
- net defined by OOA [i] based on OOA(3231, 524288, S32, 8, 8), using
- OA 4-folding and stacking [i] based on OA(3231, 2097152, S32, 8), using
- discarding factors based on OA(3231, 2097155, S32, 8), using
- discarding parts of the base [i] based on linear OA(12822, 2097155, F128, 8) (dual of [2097155, 2097133, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(12822, 2097152, F128, 8) (dual of [2097152, 2097130, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(12819, 2097152, F128, 7) (dual of [2097152, 2097133, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- discarding parts of the base [i] based on linear OA(12822, 2097155, F128, 8) (dual of [2097155, 2097133, 9]-code), using
- discarding factors based on OA(3231, 2097155, S32, 8), using
- OA 4-folding and stacking [i] based on OA(3231, 2097152, S32, 8), using
(23, 31, 1048590)-Net over F32 — Digital
Digital (23, 31, 1048590)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3231, 1048590, F32, 8) (dual of [1048590, 1048559, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- linear OA(3229, 1048576, F32, 8) (dual of [1048576, 1048547, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(3217, 1048576, F32, 5) (dual of [1048576, 1048559, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(7) ⊂ Ce(4) [i] based on
(23, 31, large)-Net in Base 32 — Upper bound on s
There is no (23, 31, large)-net in base 32, because
- 6 times m-reduction [i] would yield (23, 25, large)-net in base 32, but