Best Known (31−15, 31, s)-Nets in Base 128
(31−15, 31, 2341)-Net over F128 — Constructive and digital
Digital (16, 31, 2341)-net over F128, using
- 1281 times duplication [i] based on digital (15, 30, 2341)-net over F128, using
- net defined by OOA [i] based on linear OOA(12830, 2341, F128, 15, 15) (dual of [(2341, 15), 35085, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12830, 16388, F128, 15) (dual of [16388, 16358, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(12830, 16390, F128, 15) (dual of [16390, 16360, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(12829, 16385, F128, 15) (dual of [16385, 16356, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- 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(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([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12830, 16390, F128, 15) (dual of [16390, 16360, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12830, 16388, F128, 15) (dual of [16388, 16358, 16]-code), using
- net defined by OOA [i] based on linear OOA(12830, 2341, F128, 15, 15) (dual of [(2341, 15), 35085, 16]-NRT-code), using
(31−15, 31, 5464)-Net over F128 — Digital
Digital (16, 31, 5464)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12831, 5464, F128, 3, 15) (dual of [(5464, 3), 16361, 16]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12831, 16392, F128, 15) (dual of [16392, 16361, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(12829, 16384, F128, 15) (dual of [16384, 16355, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(12823, 16384, F128, 12) (dual of [16384, 16361, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(14) ⊂ Ce(11) [i] based on
- OOA 3-folding [i] based on linear OA(12831, 16392, F128, 15) (dual of [16392, 16361, 16]-code), using
(31−15, 31, large)-Net in Base 128 — Upper bound on s
There is no (16, 31, large)-net in base 128, because
- 13 times m-reduction [i] would yield (16, 18, large)-net in base 128, but