Best Known (61−30, 61, s)-Nets in Base 128
(61−30, 61, 1092)-Net over F128 — Constructive and digital
Digital (31, 61, 1092)-net over F128, using
- t-expansion [i] based on digital (30, 61, 1092)-net over F128, using
- net defined by OOA [i] based on linear OOA(12861, 1092, F128, 31, 31) (dual of [(1092, 31), 33791, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(12861, 16381, F128, 31) (dual of [16381, 16320, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(12861, 16384, F128, 31) (dual of [16384, 16323, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- discarding factors / shortening the dual code based on linear OA(12861, 16384, F128, 31) (dual of [16384, 16323, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(12861, 16381, F128, 31) (dual of [16381, 16320, 32]-code), using
- net defined by OOA [i] based on linear OOA(12861, 1092, F128, 31, 31) (dual of [(1092, 31), 33791, 32]-NRT-code), using
(61−30, 61, 4158)-Net over F128 — Digital
Digital (31, 61, 4158)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12861, 4158, F128, 3, 30) (dual of [(4158, 3), 12413, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12861, 5464, F128, 3, 30) (dual of [(5464, 3), 16331, 31]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12861, 16392, F128, 30) (dual of [16392, 16331, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(26) [i] based on
- linear OA(12859, 16384, F128, 30) (dual of [16384, 16325, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(12853, 16384, F128, 27) (dual of [16384, 16331, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(29) ⊂ Ce(26) [i] based on
- OOA 3-folding [i] based on linear OA(12861, 16392, F128, 30) (dual of [16392, 16331, 31]-code), using
- discarding factors / shortening the dual code based on linear OOA(12861, 5464, F128, 3, 30) (dual of [(5464, 3), 16331, 31]-NRT-code), using
(61−30, 61, large)-Net in Base 128 — Upper bound on s
There is no (31, 61, large)-net in base 128, because
- 28 times m-reduction [i] would yield (31, 33, large)-net in base 128, but