Best Known (41, 41+37, s)-Nets in Base 128
(41, 41+37, 911)-Net over F128 — Constructive and digital
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
(41, 41+37, 5467)-Net over F128 — Digital
Digital (41, 78, 5467)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12878, 5467, F128, 3, 37) (dual of [(5467, 3), 16323, 38]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12878, 16401, F128, 37) (dual of [16401, 16323, 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 3-folding [i] based on linear OA(12878, 16401, F128, 37) (dual of [16401, 16323, 38]-code), using
(41, 41+37, large)-Net in Base 128 — Upper bound on s
There is no (41, 78, large)-net in base 128, because
- 35 times m-reduction [i] would yield (41, 43, large)-net in base 128, but