Best Known (233−23, 233, s)-Nets in Base 2
(233−23, 233, 190652)-Net over F2 — Constructive and digital
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
(233−23, 233, 262146)-Net over F2 — Digital
Digital (210, 233, 262146)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2233, 262146, F2, 8, 23) (dual of [(262146, 8), 2096935, 24]-NRT-code), using
- OOA 8-folding [i] based on linear OA(2233, 2097168, F2, 23) (dual of [2097168, 2096935, 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 8-folding [i] based on linear OA(2233, 2097168, F2, 23) (dual of [2097168, 2096935, 24]-code), using
(233−23, 233, large)-Net in Base 2 — Upper bound on s
There is no (210, 233, large)-net in base 2, because
- 21 times m-reduction [i] would yield (210, 212, large)-net in base 2, but