Best Known (77−36, 77, s)-Nets in Base 128
(77−36, 77, 911)-Net over F128 — Constructive and digital
Digital (41, 77, 911)-net over F128, using
- 1 times m-reduction [i] based on digital (41, 78, 911)-net over F128, using
- net defined by OOA [i] based on linear OOA(12878, 911, F128, 37, 37) (dual of [(911, 37), 33629, 38]-NRT-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(12878, 16399, F128, 37) (dual of [16399, 16321, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(12878, 16402, F128, 37) (dual of [16402, 16324, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,15]) [i] based on
- linear OA(12873, 16385, F128, 37) (dual of [16385, 16312, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(12861, 16385, F128, 31) (dual of [16385, 16324, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (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([0,18]) ⊂ C([0,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12878, 16402, F128, 37) (dual of [16402, 16324, 38]-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(12878, 16399, F128, 37) (dual of [16399, 16321, 38]-code), using
- net defined by OOA [i] based on linear OOA(12878, 911, F128, 37, 37) (dual of [(911, 37), 33629, 38]-NRT-code), using
(77−36, 77, 6362)-Net over F128 — Digital
Digital (41, 77, 6362)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12877, 6362, F128, 2, 36) (dual of [(6362, 2), 12647, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12877, 8202, F128, 2, 36) (dual of [(8202, 2), 16327, 37]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12877, 16404, F128, 36) (dual of [16404, 16327, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(28) [i] based on
- linear OA(12871, 16384, F128, 36) (dual of [16384, 16313, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(12857, 16384, F128, 29) (dual of [16384, 16327, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(35) ⊂ Ce(28) [i] based on
- OOA 2-folding [i] based on linear OA(12877, 16404, F128, 36) (dual of [16404, 16327, 37]-code), using
- discarding factors / shortening the dual code based on linear OOA(12877, 8202, F128, 2, 36) (dual of [(8202, 2), 16327, 37]-NRT-code), using
(77−36, 77, large)-Net in Base 128 — Upper bound on s
There is no (41, 77, large)-net in base 128, because
- 34 times m-reduction [i] would yield (41, 43, large)-net in base 128, but