Best Known (212, 237, s)-Nets in Base 2
(212, 237, 43694)-Net over F2 — Constructive and digital
Digital (212, 237, 43694)-net over F2, using
- 21 times duplication [i] based on digital (211, 236, 43694)-net over F2, using
- net defined by OOA [i] based on linear OOA(2236, 43694, F2, 25, 25) (dual of [(43694, 25), 1092114, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2236, 524329, F2, 25) (dual of [524329, 524093, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2236, 524334, F2, 25) (dual of [524334, 524098, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(2229, 524289, F2, 25) (dual of [524289, 524060, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 524289 | 238−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2191, 524289, F2, 21) (dual of [524289, 524098, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 524289 | 238−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(27, 45, F2, 3) (dual of [45, 38, 4]-code or 45-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([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2236, 524334, F2, 25) (dual of [524334, 524098, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2236, 524329, F2, 25) (dual of [524329, 524093, 26]-code), using
- net defined by OOA [i] based on linear OOA(2236, 43694, F2, 25, 25) (dual of [(43694, 25), 1092114, 26]-NRT-code), using
(212, 237, 74905)-Net over F2 — Digital
Digital (212, 237, 74905)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2237, 74905, F2, 7, 25) (dual of [(74905, 7), 524098, 26]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2237, 524335, F2, 25) (dual of [524335, 524098, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2236, 524334, F2, 25) (dual of [524334, 524098, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(2229, 524289, F2, 25) (dual of [524289, 524060, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 524289 | 238−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2191, 524289, F2, 21) (dual of [524289, 524098, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 524289 | 238−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(27, 45, F2, 3) (dual of [45, 38, 4]-code or 45-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([0,12]) ⊂ C([0,10]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2236, 524334, F2, 25) (dual of [524334, 524098, 26]-code), using
- OOA 7-folding [i] based on linear OA(2237, 524335, F2, 25) (dual of [524335, 524098, 26]-code), using
(212, 237, 4401657)-Net in Base 2 — Upper bound on s
There is no (212, 237, 4401658)-net in base 2, because
- 1 times m-reduction [i] would yield (212, 236, 4401658)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 110428 024856 815107 923606 612119 023569 555024 339398 866586 194822 021716 626164 > 2236 [i]