Best Known (26, 26+25, s)-Nets in Base 128
(26, 26+25, 1365)-Net over F128 — Constructive and digital
Digital (26, 51, 1365)-net over F128, using
- 1282 times duplication [i] based on digital (24, 49, 1365)-net over F128, using
- net defined by OOA [i] based on linear OOA(12849, 1365, F128, 25, 25) (dual of [(1365, 25), 34076, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12849, 16381, F128, 25) (dual of [16381, 16332, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(12849, 16384, F128, 25) (dual of [16384, 16335, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(12849, 16384, F128, 25) (dual of [16384, 16335, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12849, 16381, F128, 25) (dual of [16381, 16332, 26]-code), using
- net defined by OOA [i] based on linear OOA(12849, 1365, F128, 25, 25) (dual of [(1365, 25), 34076, 26]-NRT-code), using
(26, 26+25, 4469)-Net over F128 — Digital
Digital (26, 51, 4469)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12851, 4469, F128, 3, 25) (dual of [(4469, 3), 13356, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12851, 5464, F128, 3, 25) (dual of [(5464, 3), 16341, 26]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12851, 16392, F128, 25) (dual of [16392, 16341, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(12849, 16384, F128, 25) (dual of [16384, 16335, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(12843, 16384, F128, 22) (dual of [16384, 16341, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(24) ⊂ Ce(21) [i] based on
- OOA 3-folding [i] based on linear OA(12851, 16392, F128, 25) (dual of [16392, 16341, 26]-code), using
- discarding factors / shortening the dual code based on linear OOA(12851, 5464, F128, 3, 25) (dual of [(5464, 3), 16341, 26]-NRT-code), using
(26, 26+25, large)-Net in Base 128 — Upper bound on s
There is no (26, 51, large)-net in base 128, because
- 23 times m-reduction [i] would yield (26, 28, large)-net in base 128, but