Best Known (117−19, 117, s)-Nets in Base 2
(117−19, 117, 458)-Net over F2 — Constructive and digital
Digital (98, 117, 458)-net over F2, using
- 22 times duplication [i] based on digital (96, 115, 458)-net over F2, using
- net defined by OOA [i] based on linear OOA(2115, 458, F2, 19, 19) (dual of [(458, 19), 8587, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2115, 4123, F2, 19) (dual of [4123, 4008, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2115, 4126, F2, 19) (dual of [4126, 4011, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(2109, 4096, F2, 19) (dual of [4096, 3987, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(285, 4096, F2, 15) (dual of [4096, 4011, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(26, 30, F2, 3) (dual of [30, 24, 4]-code or 30-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(2115, 4126, F2, 19) (dual of [4126, 4011, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2115, 4123, F2, 19) (dual of [4123, 4008, 20]-code), using
- net defined by OOA [i] based on linear OOA(2115, 458, F2, 19, 19) (dual of [(458, 19), 8587, 20]-NRT-code), using
(117−19, 117, 1225)-Net over F2 — Digital
Digital (98, 117, 1225)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2117, 1225, F2, 3, 19) (dual of [(1225, 3), 3558, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2117, 1376, F2, 3, 19) (dual of [(1376, 3), 4011, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2117, 4128, F2, 19) (dual of [4128, 4011, 20]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2115, 4126, F2, 19) (dual of [4126, 4011, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(2109, 4096, F2, 19) (dual of [4096, 3987, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(285, 4096, F2, 15) (dual of [4096, 4011, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(26, 30, F2, 3) (dual of [30, 24, 4]-code or 30-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2115, 4126, F2, 19) (dual of [4126, 4011, 20]-code), using
- OOA 3-folding [i] based on linear OA(2117, 4128, F2, 19) (dual of [4128, 4011, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(2117, 1376, F2, 3, 19) (dual of [(1376, 3), 4011, 20]-NRT-code), using
(117−19, 117, 31442)-Net in Base 2 — Upper bound on s
There is no (98, 117, 31443)-net in base 2, because
- 1 times m-reduction [i] would yield (98, 116, 31443)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 83093 707134 516147 953000 241411 331080 > 2116 [i]