Best Known (211, 211+22, s)-Nets in Base 2
(211, 211+22, 190652)-Net over F2 — Constructive and digital
Digital (211, 233, 190652)-net over F2, using
- t-expansion [i] based on digital (210, 233, 190652)-net over F2, using
- net defined by OOA [i] based on linear OOA(2233, 190652, F2, 23, 23) (dual of [(190652, 23), 4384763, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2233, 2097173, F2, 23) (dual of [2097173, 2096940, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2233, 2097174, F2, 23) (dual of [2097174, 2096941, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(2232, 2097152, F2, 23) (dual of [2097152, 2096920, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2211, 2097152, F2, 21) (dual of [2097152, 2096941, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(21, 22, F2, 1) (dual of [22, 21, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(2233, 2097174, F2, 23) (dual of [2097174, 2096941, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2233, 2097173, F2, 23) (dual of [2097173, 2096940, 24]-code), using
- net defined by OOA [i] based on linear OOA(2233, 190652, F2, 23, 23) (dual of [(190652, 23), 4384763, 24]-NRT-code), using
(211, 211+22, 299596)-Net over F2 — Digital
Digital (211, 233, 299596)-net over F2, using
- 21 times duplication [i] based on digital (210, 232, 299596)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2232, 299596, F2, 7, 22) (dual of [(299596, 7), 2096940, 23]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2232, 2097172, F2, 22) (dual of [2097172, 2096940, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(2232, 2097173, F2, 22) (dual of [2097173, 2096941, 23]-code), using
- 1 times truncation [i] based on linear OA(2233, 2097174, F2, 23) (dual of [2097174, 2096941, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(2232, 2097152, F2, 23) (dual of [2097152, 2096920, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2211, 2097152, F2, 21) (dual of [2097152, 2096941, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(21, 22, F2, 1) (dual of [22, 21, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- 1 times truncation [i] based on linear OA(2233, 2097174, F2, 23) (dual of [2097174, 2096941, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2232, 2097173, F2, 22) (dual of [2097173, 2096941, 23]-code), using
- OOA 7-folding [i] based on linear OA(2232, 2097172, F2, 22) (dual of [2097172, 2096940, 23]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2232, 299596, F2, 7, 22) (dual of [(299596, 7), 2096940, 23]-NRT-code), using
(211, 211+22, large)-Net in Base 2 — Upper bound on s
There is no (211, 233, large)-net in base 2, because
- 20 times m-reduction [i] would yield (211, 213, large)-net in base 2, but