Best Known (14, 14+13, s)-Nets in Base 128
(14, 14+13, 2731)-Net over F128 — Constructive and digital
Digital (14, 27, 2731)-net over F128, using
- 1281 times duplication [i] based on digital (13, 26, 2731)-net over F128, using
- net defined by OOA [i] based on linear OOA(12826, 2731, F128, 13, 13) (dual of [(2731, 13), 35477, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12826, 16387, F128, 13) (dual of [16387, 16361, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(12826, 16390, F128, 13) (dual of [16390, 16364, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(12825, 16385, F128, 13) (dual of [16385, 16360, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(12821, 16385, F128, 11) (dual of [16385, 16364, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(1281, 5, F128, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12826, 16390, F128, 13) (dual of [16390, 16364, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12826, 16387, F128, 13) (dual of [16387, 16361, 14]-code), using
- net defined by OOA [i] based on linear OOA(12826, 2731, F128, 13, 13) (dual of [(2731, 13), 35477, 14]-NRT-code), using
(14, 14+13, 6606)-Net over F128 — Digital
Digital (14, 27, 6606)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12827, 6606, F128, 2, 13) (dual of [(6606, 2), 13185, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12827, 8196, F128, 2, 13) (dual of [(8196, 2), 16365, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12827, 16392, F128, 13) (dual of [16392, 16365, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(12825, 16384, F128, 13) (dual of [16384, 16359, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(12819, 16384, F128, 10) (dual of [16384, 16365, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1282, 8, F128, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- OOA 2-folding [i] based on linear OA(12827, 16392, F128, 13) (dual of [16392, 16365, 14]-code), using
- discarding factors / shortening the dual code based on linear OOA(12827, 8196, F128, 2, 13) (dual of [(8196, 2), 16365, 14]-NRT-code), using
(14, 14+13, large)-Net in Base 128 — Upper bound on s
There is no (14, 27, large)-net in base 128, because
- 11 times m-reduction [i] would yield (14, 16, large)-net in base 128, but