Best Known (18, 18+17, s)-Nets in Base 128
(18, 18+17, 2048)-Net over F128 — Constructive and digital
Digital (18, 35, 2048)-net over F128, using
- 1282 times duplication [i] based on digital (16, 33, 2048)-net over F128, using
- net defined by OOA [i] based on linear OOA(12833, 2048, F128, 17, 17) (dual of [(2048, 17), 34783, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(12833, 16385, F128, 17) (dual of [16385, 16352, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(12833, 16385, F128, 17) (dual of [16385, 16352, 18]-code), using
- net defined by OOA [i] based on linear OOA(12833, 2048, F128, 17, 17) (dual of [(2048, 17), 34783, 18]-NRT-code), using
(18, 18+17, 5464)-Net over F128 — Digital
Digital (18, 35, 5464)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12835, 5464, F128, 3, 17) (dual of [(5464, 3), 16357, 18]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12835, 16392, F128, 17) (dual of [16392, 16357, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(12833, 16384, F128, 17) (dual of [16384, 16351, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(12827, 16384, F128, 14) (dual of [16384, 16357, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(16) ⊂ Ce(13) [i] based on
- OOA 3-folding [i] based on linear OA(12835, 16392, F128, 17) (dual of [16392, 16357, 18]-code), using
(18, 18+17, large)-Net in Base 128 — Upper bound on s
There is no (18, 35, large)-net in base 128, because
- 15 times m-reduction [i] would yield (18, 20, large)-net in base 128, but