Best Known (37, 37+36, s)-Nets in Base 128
(37, 37+36, 910)-Net over F128 — Constructive and digital
Digital (37, 73, 910)-net over F128, using
- t-expansion [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
(37, 37+36, 4098)-Net over F128 — Digital
Digital (37, 73, 4098)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12873, 4098, F128, 4, 36) (dual of [(4098, 4), 16319, 37]-NRT-code), using
- OOA 4-folding [i] based on linear OA(12873, 16392, F128, 36) (dual of [16392, 16319, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(32) [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(12865, 16384, F128, 33) (dual of [16384, 16319, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [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(35) ⊂ Ce(32) [i] based on
- OOA 4-folding [i] based on linear OA(12873, 16392, F128, 36) (dual of [16392, 16319, 37]-code), using
(37, 37+36, large)-Net in Base 128 — Upper bound on s
There is no (37, 73, large)-net in base 128, because
- 34 times m-reduction [i] would yield (37, 39, large)-net in base 128, but