Best Known (38, 38+37, s)-Nets in Base 128
(38, 38+37, 910)-Net over F128 — Constructive and digital
Digital (38, 75, 910)-net over F128, using
- 1282 times duplication [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
(38, 38+37, 4098)-Net over F128 — Digital
Digital (38, 75, 4098)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12875, 4098, F128, 4, 37) (dual of [(4098, 4), 16317, 38]-NRT-code), using
- OOA 4-folding [i] based on linear OA(12875, 16392, F128, 37) (dual of [16392, 16317, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(33) [i] 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]
- 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(1282, 8, F128, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(36) ⊂ Ce(33) [i] based on
- OOA 4-folding [i] based on linear OA(12875, 16392, F128, 37) (dual of [16392, 16317, 38]-code), using
(38, 38+37, large)-Net in Base 128 — Upper bound on s
There is no (38, 75, large)-net in base 128, because
- 35 times m-reduction [i] would yield (38, 40, large)-net in base 128, but