Best Known (72−23, 72, s)-Nets in Base 128
(72−23, 72, 190652)-Net over F128 — Constructive and digital
Digital (49, 72, 190652)-net over F128, using
- net defined by OOA [i] based on linear OOA(12872, 190652, F128, 23, 23) (dual of [(190652, 23), 4384924, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(12872, 2097173, F128, 23) (dual of [2097173, 2097101, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(12872, 2097176, F128, 23) (dual of [2097176, 2097104, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- linear OA(12867, 2097153, F128, 23) (dual of [2097153, 2097086, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(12849, 2097153, F128, 17) (dual of [2097153, 2097104, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(1285, 23, F128, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,128)), using
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- Reed–Solomon code RS(123,128) [i]
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12872, 2097176, F128, 23) (dual of [2097176, 2097104, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(12872, 2097173, F128, 23) (dual of [2097173, 2097101, 24]-code), using
(72−23, 72, 1048588)-Net over F128 — Digital
Digital (49, 72, 1048588)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12872, 1048588, F128, 2, 23) (dual of [(1048588, 2), 2097104, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12872, 2097176, F128, 23) (dual of [2097176, 2097104, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- linear OA(12867, 2097153, F128, 23) (dual of [2097153, 2097086, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(12849, 2097153, F128, 17) (dual of [2097153, 2097104, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(1285, 23, F128, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,128)), using
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- Reed–Solomon code RS(123,128) [i]
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- OOA 2-folding [i] based on linear OA(12872, 2097176, F128, 23) (dual of [2097176, 2097104, 24]-code), using
(72−23, 72, large)-Net in Base 128 — Upper bound on s
There is no (49, 72, large)-net in base 128, because
- 21 times m-reduction [i] would yield (49, 51, large)-net in base 128, but