Best Known (71, 71+23, s)-Nets in Base 32
(71, 71+23, 95327)-Net over F32 — Constructive and digital
Digital (71, 94, 95327)-net over F32, using
- 321 times duplication [i] based on digital (70, 93, 95327)-net over F32, using
- net defined by OOA [i] based on linear OOA(3293, 95327, F32, 23, 23) (dual of [(95327, 23), 2192428, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3293, 1048598, F32, 23) (dual of [1048598, 1048505, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3293, 1048600, F32, 23) (dual of [1048600, 1048507, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- linear OA(3289, 1048576, F32, 23) (dual of [1048576, 1048487, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3269, 1048576, F32, 18) (dual of [1048576, 1048507, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(22) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(3293, 1048600, F32, 23) (dual of [1048600, 1048507, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3293, 1048598, F32, 23) (dual of [1048598, 1048505, 24]-code), using
- net defined by OOA [i] based on linear OOA(3293, 95327, F32, 23, 23) (dual of [(95327, 23), 2192428, 24]-NRT-code), using
(71, 71+23, 190650)-Net in Base 32 — Constructive
(71, 94, 190650)-net in base 32, using
- net defined by OOA [i] based on OOA(3294, 190650, S32, 23, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(3294, 2097151, S32, 23), using
- discarding factors based on OA(3294, 2097155, S32, 23), using
- discarding parts of the base [i] based on linear OA(12867, 2097155, F128, 23) (dual of [2097155, 2097088, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(12867, 2097152, F128, 23) (dual of [2097152, 2097085, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 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(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(22) ⊂ Ce(21) [i] based on
- discarding parts of the base [i] based on linear OA(12867, 2097155, F128, 23) (dual of [2097155, 2097088, 24]-code), using
- discarding factors based on OA(3294, 2097155, S32, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(3294, 2097151, S32, 23), using
(71, 71+23, 1048606)-Net over F32 — Digital
Digital (71, 94, 1048606)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3294, 1048606, F32, 23) (dual of [1048606, 1048512, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- linear OA(3289, 1048577, F32, 23) (dual of [1048577, 1048488, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(3265, 1048577, F32, 17) (dual of [1048577, 1048512, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [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 C([0,11]) ⊂ C([0,8]) [i] based on
(71, 71+23, large)-Net in Base 32 — Upper bound on s
There is no (71, 94, large)-net in base 32, because
- 21 times m-reduction [i] would yield (71, 73, large)-net in base 32, but