Best Known (68, 68+22, s)-Nets in Base 32
(68, 68+22, 95327)-Net over F32 — Constructive and digital
Digital (68, 90, 95327)-net over F32, using
- 321 times duplication [i] based on digital (67, 89, 95327)-net over F32, using
- net defined by OOA [i] based on linear OOA(3289, 95327, F32, 22, 22) (dual of [(95327, 22), 2097105, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(3289, 1048597, F32, 22) (dual of [1048597, 1048508, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3289, 1048600, F32, 22) (dual of [1048600, 1048511, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(3285, 1048576, F32, 22) (dual of [1048576, 1048491, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3265, 1048576, F32, 17) (dual of [1048576, 1048511, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(324, 24, F32, 4) (dual of [24, 20, 5]-code or 24-arc in PG(3,32)), using
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- Reed–Solomon code RS(28,32) [i]
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3289, 1048600, F32, 22) (dual of [1048600, 1048511, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(3289, 1048597, F32, 22) (dual of [1048597, 1048508, 23]-code), using
- net defined by OOA [i] based on linear OOA(3289, 95327, F32, 22, 22) (dual of [(95327, 22), 2097105, 23]-NRT-code), using
(68, 68+22, 190650)-Net in Base 32 — Constructive
(68, 90, 190650)-net in base 32, using
- net defined by OOA [i] based on OOA(3290, 190650, S32, 22, 22), using
- OA 11-folding and stacking [i] based on OA(3290, 2097150, S32, 22), using
- discarding factors based on OA(3290, 2097155, S32, 22), using
- discarding parts of the base [i] based on linear OA(12864, 2097155, F128, 22) (dual of [2097155, 2097091, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(12864, 2097152, F128, 22) (dual of [2097152, 2097088, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(12861, 2097152, F128, 21) (dual of [2097152, 2097091, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(21) ⊂ Ce(20) [i] based on
- discarding parts of the base [i] based on linear OA(12864, 2097155, F128, 22) (dual of [2097155, 2097091, 23]-code), using
- discarding factors based on OA(3290, 2097155, S32, 22), using
- OA 11-folding and stacking [i] based on OA(3290, 2097150, S32, 22), using
(68, 68+22, 1048605)-Net over F32 — Digital
Digital (68, 90, 1048605)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3290, 1048605, F32, 22) (dual of [1048605, 1048515, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- linear OA(3285, 1048576, F32, 22) (dual of [1048576, 1048491, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3261, 1048576, F32, 16) (dual of [1048576, 1048515, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(325, 29, F32, 5) (dual of [29, 24, 6]-code or 29-arc in PG(4,32)), using
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- Reed–Solomon code RS(27,32) [i]
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
(68, 68+22, large)-Net in Base 32 — Upper bound on s
There is no (68, 90, large)-net in base 32, because
- 20 times m-reduction [i] would yield (68, 70, large)-net in base 32, but