Best Known (203, 203+28, s)-Nets in Base 2
(203, 203+28, 4683)-Net over F2 — Constructive and digital
Digital (203, 231, 4683)-net over F2, using
- t-expansion [i] based on digital (202, 231, 4683)-net over F2, using
- net defined by OOA [i] based on linear OOA(2231, 4683, F2, 29, 29) (dual of [(4683, 29), 135576, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2231, 65563, F2, 29) (dual of [65563, 65332, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2231, 65569, F2, 29) (dual of [65569, 65338, 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(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2231, 65569, F2, 29) (dual of [65569, 65338, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2231, 65563, F2, 29) (dual of [65563, 65332, 30]-code), using
- net defined by OOA [i] based on linear OOA(2231, 4683, F2, 29, 29) (dual of [(4683, 29), 135576, 30]-NRT-code), using
(203, 203+28, 11169)-Net over F2 — Digital
Digital (203, 231, 11169)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2231, 11169, F2, 5, 28) (dual of [(11169, 5), 55614, 29]-NRT-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OOA(2231, 13115, F2, 5, 28) (dual of [(13115, 5), 65344, 29]-NRT-code), using
(203, 203+28, 560320)-Net in Base 2 — Upper bound on s
There is no (203, 231, 560321)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 3450 886546 514085 046629 052013 334564 933413 564184 982420 434747 549556 146336 > 2231 [i]