Best Known (53, 53+25, s)-Nets in Base 128
(53, 53+25, 174764)-Net over F128 — Constructive and digital
Digital (53, 78, 174764)-net over F128, using
- 1281 times duplication [i] based on digital (52, 77, 174764)-net over F128, using
- net defined by OOA [i] based on linear OOA(12877, 174764, F128, 25, 25) (dual of [(174764, 25), 4369023, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12877, 2097169, F128, 25) (dual of [2097169, 2097092, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(12877, 2097171, F128, 25) (dual of [2097171, 2097094, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- linear OA(12873, 2097152, F128, 25) (dual of [2097152, 2097079, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(12858, 2097152, F128, 20) (dual of [2097152, 2097094, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(24) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(12877, 2097171, F128, 25) (dual of [2097171, 2097094, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12877, 2097169, F128, 25) (dual of [2097169, 2097092, 26]-code), using
- net defined by OOA [i] based on linear OOA(12877, 174764, F128, 25, 25) (dual of [(174764, 25), 4369023, 26]-NRT-code), using
(53, 53+25, 1048588)-Net over F128 — Digital
Digital (53, 78, 1048588)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12878, 1048588, F128, 2, 25) (dual of [(1048588, 2), 2097098, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12878, 2097176, F128, 25) (dual of [2097176, 2097098, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(12873, 2097153, F128, 25) (dual of [2097153, 2097080, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 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(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([0,12]) ⊂ C([0,9]) [i] based on
- OOA 2-folding [i] based on linear OA(12878, 2097176, F128, 25) (dual of [2097176, 2097098, 26]-code), using
(53, 53+25, large)-Net in Base 128 — Upper bound on s
There is no (53, 78, large)-net in base 128, because
- 23 times m-reduction [i] would yield (53, 55, large)-net in base 128, but