Best Known (37, 54, s)-Nets in Base 128
(37, 54, 262146)-Net over F128 — Constructive and digital
Digital (37, 54, 262146)-net over F128, using
- 1281 times duplication [i] based on digital (36, 53, 262146)-net over F128, using
- net defined by OOA [i] based on linear OOA(12853, 262146, F128, 17, 17) (dual of [(262146, 17), 4456429, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(12853, 2097169, F128, 17) (dual of [2097169, 2097116, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(12853, 2097171, F128, 17) (dual of [2097171, 2097118, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- linear OA(12849, 2097152, F128, 17) (dual of [2097152, 2097103, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(1284, 19, F128, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,128)), using
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- Reed–Solomon code RS(124,128) [i]
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(12853, 2097171, F128, 17) (dual of [2097171, 2097118, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(12853, 2097169, F128, 17) (dual of [2097169, 2097116, 18]-code), using
- net defined by OOA [i] based on linear OOA(12853, 262146, F128, 17, 17) (dual of [(262146, 17), 4456429, 18]-NRT-code), using
(37, 54, 1410705)-Net over F128 — Digital
Digital (37, 54, 1410705)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12854, 1410705, F128, 17) (dual of [1410705, 1410651, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(12854, 2097176, F128, 17) (dual of [2097176, 2097122, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- 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(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(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([0,8]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12854, 2097176, F128, 17) (dual of [2097176, 2097122, 18]-code), using
(37, 54, large)-Net in Base 128 — Upper bound on s
There is no (37, 54, large)-net in base 128, because
- 15 times m-reduction [i] would yield (37, 39, large)-net in base 128, but