Best Known (72−36, 72, s)-Nets in Base 128
(72−36, 72, 910)-Net over F128 — Constructive and digital
Digital (36, 72, 910)-net over F128, using
- 1 times m-reduction [i] based on digital (36, 73, 910)-net over F128, using
- net defined by OOA [i] based on linear OOA(12873, 910, F128, 37, 37) (dual of [(910, 37), 33597, 38]-NRT-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(12873, 16381, F128, 37) (dual of [16381, 16308, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(12873, 16384, F128, 37) (dual of [16384, 16311, 38]-code), using
- an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- discarding factors / shortening the dual code based on linear OA(12873, 16384, F128, 37) (dual of [16384, 16311, 38]-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(12873, 16381, F128, 37) (dual of [16381, 16308, 38]-code), using
- net defined by OOA [i] based on linear OOA(12873, 910, F128, 37, 37) (dual of [(910, 37), 33597, 38]-NRT-code), using
(72−36, 72, 4083)-Net over F128 — Digital
Digital (36, 72, 4083)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12872, 4083, F128, 4, 36) (dual of [(4083, 4), 16260, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12872, 4097, F128, 4, 36) (dual of [(4097, 4), 16316, 37]-NRT-code), using
- OOA 4-folding [i] based on linear OA(12872, 16388, F128, 36) (dual of [16388, 16316, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(12872, 16389, F128, 36) (dual of [16389, 16317, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(33) [i] based on
- linear OA(12871, 16384, F128, 36) (dual of [16384, 16313, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(12867, 16384, F128, 34) (dual of [16384, 16317, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(1281, 5, F128, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(35) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(12872, 16389, F128, 36) (dual of [16389, 16317, 37]-code), using
- OOA 4-folding [i] based on linear OA(12872, 16388, F128, 36) (dual of [16388, 16316, 37]-code), using
- discarding factors / shortening the dual code based on linear OOA(12872, 4097, F128, 4, 36) (dual of [(4097, 4), 16316, 37]-NRT-code), using
(72−36, 72, large)-Net in Base 128 — Upper bound on s
There is no (36, 72, large)-net in base 128, because
- 34 times m-reduction [i] would yield (36, 38, large)-net in base 128, but