Best Known (234−28, 234, s)-Nets in Base 2
(234−28, 234, 4684)-Net over F2 — Constructive and digital
Digital (206, 234, 4684)-net over F2, using
- 21 times duplication [i] based on digital (205, 233, 4684)-net over F2, using
- t-expansion [i] based on digital (204, 233, 4684)-net over F2, using
- net defined by OOA [i] based on linear OOA(2233, 4684, F2, 29, 29) (dual of [(4684, 29), 135603, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2233, 65577, F2, 29) (dual of [65577, 65344, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2232, 65576, F2, 29) (dual of [65576, 65344, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(2225, 65537, F2, 29) (dual of [65537, 65312, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(2193, 65537, F2, 25) (dual of [65537, 65344, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(27, 39, F2, 3) (dual of [39, 32, 4]-code or 39-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2232, 65576, F2, 29) (dual of [65576, 65344, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2233, 65577, F2, 29) (dual of [65577, 65344, 30]-code), using
- net defined by OOA [i] based on linear OOA(2233, 4684, F2, 29, 29) (dual of [(4684, 29), 135603, 30]-NRT-code), using
- t-expansion [i] based on digital (204, 233, 4684)-net over F2, using
(234−28, 234, 12279)-Net over F2 — Digital
Digital (206, 234, 12279)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2234, 12279, F2, 5, 28) (dual of [(12279, 5), 61161, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2234, 13115, F2, 5, 28) (dual of [(13115, 5), 65341, 29]-NRT-code), using
- 23 times duplication [i] based on linear OOA(2231, 13115, F2, 5, 28) (dual of [(13115, 5), 65344, 29]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2231, 65575, F2, 28) (dual of [65575, 65344, 29]-code), using
- 1 times truncation [i] based on linear OA(2232, 65576, F2, 29) (dual of [65576, 65344, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(2225, 65537, F2, 29) (dual of [65537, 65312, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(2193, 65537, F2, 25) (dual of [65537, 65344, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(27, 39, F2, 3) (dual of [39, 32, 4]-code or 39-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- 1 times truncation [i] based on linear OA(2232, 65576, F2, 29) (dual of [65576, 65344, 30]-code), using
- OOA 5-folding [i] based on linear OA(2231, 65575, F2, 28) (dual of [65575, 65344, 29]-code), using
- 23 times duplication [i] based on linear OOA(2231, 13115, F2, 5, 28) (dual of [(13115, 5), 65344, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2234, 13115, F2, 5, 28) (dual of [(13115, 5), 65341, 29]-NRT-code), using
(234−28, 234, 650047)-Net in Base 2 — Upper bound on s
There is no (206, 234, 650048)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 27607 008936 495639 664504 997027 892272 308886 059710 584756 106902 087435 713689 > 2234 [i]