Best Known (63−19, 63, s)-Nets in Base 128
(63−19, 63, 233020)-Net over F128 — Constructive and digital
Digital (44, 63, 233020)-net over F128, using
- 1281 times duplication [i] based on digital (43, 62, 233020)-net over F128, using
- net defined by OOA [i] based on linear OOA(12862, 233020, F128, 19, 19) (dual of [(233020, 19), 4427318, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12862, 2097181, F128, 19) (dual of [2097181, 2097119, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(12862, 2097184, F128, 19) (dual of [2097184, 2097122, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,5]) [i] based on
- linear OA(12855, 2097153, F128, 19) (dual of [2097153, 2097098, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(12831, 2097153, F128, 11) (dual of [2097153, 2097122, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(1287, 31, F128, 7) (dual of [31, 24, 8]-code or 31-arc in PG(6,128)), using
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- Reed–Solomon code RS(121,128) [i]
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- construction X applied to C([0,9]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12862, 2097184, F128, 19) (dual of [2097184, 2097122, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12862, 2097181, F128, 19) (dual of [2097181, 2097119, 20]-code), using
- net defined by OOA [i] based on linear OOA(12862, 233020, F128, 19, 19) (dual of [(233020, 19), 4427318, 20]-NRT-code), using
(63−19, 63, 932066)-Net in Base 128 — Constructive
(44, 63, 932066)-net in base 128, using
- net defined by OOA [i] based on OOA(12863, 932066, S128, 19, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(12863, 8388595, S128, 19), using
- discarding factors based on OA(12863, large, S128, 19), using
- discarding parts of the base [i] based on linear OA(25655, large, F256, 19) (dual of [large, large−55, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding parts of the base [i] based on linear OA(25655, large, F256, 19) (dual of [large, large−55, 20]-code), using
- discarding factors based on OA(12863, large, S128, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(12863, 8388595, S128, 19), using
(63−19, 63, 2097187)-Net over F128 — Digital
Digital (44, 63, 2097187)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12863, 2097187, F128, 19) (dual of [2097187, 2097124, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(9) [i] based on
- linear OA(12855, 2097152, F128, 19) (dual of [2097152, 2097097, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(12828, 2097152, F128, 10) (dual of [2097152, 2097124, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1288, 35, F128, 8) (dual of [35, 27, 9]-code or 35-arc in PG(7,128)), using
- discarding factors / shortening the dual code based on linear OA(1288, 128, F128, 8) (dual of [128, 120, 9]-code or 128-arc in PG(7,128)), using
- Reed–Solomon code RS(120,128) [i]
- discarding factors / shortening the dual code based on linear OA(1288, 128, F128, 8) (dual of [128, 120, 9]-code or 128-arc in PG(7,128)), using
- construction X applied to Ce(18) ⊂ Ce(9) [i] based on
(63−19, 63, large)-Net in Base 128 — Upper bound on s
There is no (44, 63, large)-net in base 128, because
- 17 times m-reduction [i] would yield (44, 46, large)-net in base 128, but