Best Known (137−20, 137, s)-Nets in Base 2
(137−20, 137, 822)-Net over F2 — Constructive and digital
Digital (117, 137, 822)-net over F2, using
- t-expansion [i] based on digital (116, 137, 822)-net over F2, using
- net defined by OOA [i] based on linear OOA(2137, 822, F2, 21, 21) (dual of [(822, 21), 17125, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2137, 8221, F2, 21) (dual of [8221, 8084, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2137, 8225, F2, 21) (dual of [8225, 8088, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2131, 8193, F2, 21) (dual of [8193, 8062, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2105, 8193, F2, 17) (dual of [8193, 8088, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2137, 8225, F2, 21) (dual of [8225, 8088, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2137, 8221, F2, 21) (dual of [8221, 8084, 22]-code), using
- net defined by OOA [i] based on linear OOA(2137, 822, F2, 21, 21) (dual of [(822, 21), 17125, 22]-NRT-code), using
(137−20, 137, 2235)-Net over F2 — Digital
Digital (117, 137, 2235)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2137, 2235, F2, 3, 20) (dual of [(2235, 3), 6568, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2137, 2741, F2, 3, 20) (dual of [(2741, 3), 8086, 21]-NRT-code), using
- strength reduction [i] based on linear OOA(2137, 2741, F2, 3, 21) (dual of [(2741, 3), 8086, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2137, 8223, F2, 21) (dual of [8223, 8086, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2137, 8225, F2, 21) (dual of [8225, 8088, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2131, 8193, F2, 21) (dual of [8193, 8062, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2105, 8193, F2, 17) (dual of [8193, 8088, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2137, 8225, F2, 21) (dual of [8225, 8088, 22]-code), using
- OOA 3-folding [i] based on linear OA(2137, 8223, F2, 21) (dual of [8223, 8086, 22]-code), using
- strength reduction [i] based on linear OOA(2137, 2741, F2, 3, 21) (dual of [(2741, 3), 8086, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2137, 2741, F2, 3, 20) (dual of [(2741, 3), 8086, 21]-NRT-code), using
(137−20, 137, 60253)-Net in Base 2 — Upper bound on s
There is no (117, 137, 60254)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 174237 118704 933136 214872 581268 719499 392354 > 2137 [i]