Best Known (34, 49, s)-Nets in Base 128
(34, 49, 299596)-Net over F128 — Constructive and digital
Digital (34, 49, 299596)-net over F128, using
- 1281 times duplication [i] based on digital (33, 48, 299596)-net over F128, using
- net defined by OOA [i] based on linear OOA(12848, 299596, F128, 15, 15) (dual of [(299596, 15), 4493892, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12848, 2097173, F128, 15) (dual of [2097173, 2097125, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(12848, 2097176, F128, 15) (dual of [2097176, 2097128, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,4]) [i] based on
- linear OA(12843, 2097153, F128, 15) (dual of [2097153, 2097110, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(12825, 2097153, F128, 9) (dual of [2097153, 2097128, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (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([0,7]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12848, 2097176, F128, 15) (dual of [2097176, 2097128, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12848, 2097173, F128, 15) (dual of [2097173, 2097125, 16]-code), using
- net defined by OOA [i] based on linear OOA(12848, 299596, F128, 15, 15) (dual of [(299596, 15), 4493892, 16]-NRT-code), using
(34, 49, 2097179)-Net over F128 — Digital
Digital (34, 49, 2097179)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12849, 2097179, F128, 15) (dual of [2097179, 2097130, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(7) [i] based on
- linear OA(12843, 2097152, F128, 15) (dual of [2097152, 2097109, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(12822, 2097152, F128, 8) (dual of [2097152, 2097130, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(1286, 27, F128, 6) (dual of [27, 21, 7]-code or 27-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(14) ⊂ Ce(7) [i] based on
(34, 49, large)-Net in Base 128 — Upper bound on s
There is no (34, 49, large)-net in base 128, because
- 13 times m-reduction [i] would yield (34, 36, large)-net in base 128, but