Best Known (33−15, 33, s)-Nets in Base 128
(33−15, 33, 2342)-Net over F128 — Constructive and digital
Digital (18, 33, 2342)-net over F128, using
- 1281 times duplication [i] based on digital (17, 32, 2342)-net over F128, using
- net defined by OOA [i] based on linear OOA(12832, 2342, F128, 15, 15) (dual of [(2342, 15), 35098, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12832, 16395, F128, 15) (dual of [16395, 16363, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(12832, 16396, F128, 15) (dual of [16396, 16364, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [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(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(1283, 11, F128, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,128) or 11-cap in PG(2,128)), using
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- Reed–Solomon code RS(125,128) [i]
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12832, 16396, F128, 15) (dual of [16396, 16364, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12832, 16395, F128, 15) (dual of [16395, 16363, 16]-code), using
- net defined by OOA [i] based on linear OOA(12832, 2342, F128, 15, 15) (dual of [(2342, 15), 35098, 16]-NRT-code), using
(33−15, 33, 8199)-Net over F128 — Digital
Digital (18, 33, 8199)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12833, 8199, F128, 2, 15) (dual of [(8199, 2), 16365, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12833, 16398, F128, 15) (dual of [16398, 16365, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(9) [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(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(1284, 14, F128, 4) (dual of [14, 10, 5]-code or 14-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(14) ⊂ Ce(9) [i] based on
- OOA 2-folding [i] based on linear OA(12833, 16398, F128, 15) (dual of [16398, 16365, 16]-code), using
(33−15, 33, large)-Net in Base 128 — Upper bound on s
There is no (18, 33, large)-net in base 128, because
- 13 times m-reduction [i] would yield (18, 20, large)-net in base 128, but