Best Known (236−33, 236, s)-Nets in Base 2
(236−33, 236, 1026)-Net over F2 — Constructive and digital
Digital (203, 236, 1026)-net over F2, using
- 25 times duplication [i] based on digital (198, 231, 1026)-net over F2, using
- net defined by OOA [i] based on linear OOA(2231, 1026, F2, 33, 33) (dual of [(1026, 33), 33627, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(2231, 16417, F2, 33) (dual of [16417, 16186, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- linear OA(2225, 16385, F2, 33) (dual of [16385, 16160, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 228−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(2197, 16385, F2, 29) (dual of [16385, 16188, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 228−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (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([0,16]) ⊂ C([0,14]) [i] based on
- OOA 16-folding and stacking with additional row [i] based on linear OA(2231, 16417, F2, 33) (dual of [16417, 16186, 34]-code), using
- net defined by OOA [i] based on linear OOA(2231, 1026, F2, 33, 33) (dual of [(1026, 33), 33627, 34]-NRT-code), using
(236−33, 236, 3485)-Net over F2 — Digital
Digital (203, 236, 3485)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2236, 3485, F2, 4, 33) (dual of [(3485, 4), 13704, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2236, 4106, F2, 4, 33) (dual of [(4106, 4), 16188, 34]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2232, 4105, F2, 4, 33) (dual of [(4105, 4), 16188, 34]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2232, 16420, F2, 33) (dual of [16420, 16188, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- linear OA(2225, 16385, F2, 33) (dual of [16385, 16160, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 228−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(2197, 16385, F2, 29) (dual of [16385, 16188, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 228−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(27, 35, F2, 3) (dual of [35, 28, 4]-code or 35-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([0,16]) ⊂ C([0,14]) [i] based on
- OOA 4-folding [i] based on linear OA(2232, 16420, F2, 33) (dual of [16420, 16188, 34]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2232, 4105, F2, 4, 33) (dual of [(4105, 4), 16188, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2236, 4106, F2, 4, 33) (dual of [(4106, 4), 16188, 34]-NRT-code), using
(236−33, 236, 179415)-Net in Base 2 — Upper bound on s
There is no (203, 236, 179416)-net in base 2, because
- 1 times m-reduction [i] would yield (203, 235, 179416)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 55216 536168 722982 340796 722310 057389 734090 886654 262088 136089 587465 534521 > 2235 [i]