Best Known (28, 51, s)-Nets in Base 128
(28, 51, 1491)-Net over F128 — Constructive and digital
Digital (28, 51, 1491)-net over F128, using
- 1281 times duplication [i] based on digital (27, 50, 1491)-net over F128, using
- net defined by OOA [i] based on linear OOA(12850, 1491, F128, 23, 23) (dual of [(1491, 23), 34243, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(12850, 16402, F128, 23) (dual of [16402, 16352, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- linear OA(12845, 16385, F128, 23) (dual of [16385, 16340, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- 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]
- linear OA(1285, 17, F128, 5) (dual of [17, 12, 6]-code or 17-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 11-folding and stacking with additional row [i] based on linear OA(12850, 16402, F128, 23) (dual of [16402, 16352, 24]-code), using
- net defined by OOA [i] based on linear OOA(12850, 1491, F128, 23, 23) (dual of [(1491, 23), 34243, 24]-NRT-code), using
(28, 51, 8202)-Net over F128 — Digital
Digital (28, 51, 8202)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12851, 8202, F128, 2, 23) (dual of [(8202, 2), 16353, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12851, 16404, F128, 23) (dual of [16404, 16353, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(15) [i] based on
- linear OA(12845, 16384, F128, 23) (dual of [16384, 16339, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(12831, 16384, F128, 16) (dual of [16384, 16353, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(1286, 20, F128, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,128)), using
- discarding factors / shortening the dual code based on linear OA(1286, 128, F128, 6) (dual of [128, 122, 7]-code or 128-arc in PG(5,128)), using
- Reed–Solomon code RS(122,128) [i]
- discarding factors / shortening the dual code based on linear OA(1286, 128, F128, 6) (dual of [128, 122, 7]-code or 128-arc in PG(5,128)), using
- construction X applied to Ce(22) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(12851, 16404, F128, 23) (dual of [16404, 16353, 24]-code), using
(28, 51, large)-Net in Base 128 — Upper bound on s
There is no (28, 51, large)-net in base 128, because
- 21 times m-reduction [i] would yield (28, 30, large)-net in base 128, but