Best Known (234−25, 234, s)-Nets in Base 2
(234−25, 234, 43692)-Net over F2 — Constructive and digital
Digital (209, 234, 43692)-net over F2, using
- 24 times duplication [i] based on digital (205, 230, 43692)-net over F2, using
- net defined by OOA [i] based on linear OOA(2230, 43692, F2, 25, 25) (dual of [(43692, 25), 1092070, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2230, 524305, F2, 25) (dual of [524305, 524075, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2230, 524308, F2, 25) (dual of [524308, 524078, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(2229, 524288, F2, 25) (dual of [524288, 524059, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2210, 524288, F2, 23) (dual of [524288, 524078, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(21, 20, F2, 1) (dual of [20, 19, 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(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2230, 524308, F2, 25) (dual of [524308, 524078, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2230, 524305, F2, 25) (dual of [524305, 524075, 26]-code), using
- net defined by OOA [i] based on linear OOA(2230, 43692, F2, 25, 25) (dual of [(43692, 25), 1092070, 26]-NRT-code), using
(234−25, 234, 74901)-Net over F2 — Digital
Digital (209, 234, 74901)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2234, 74901, F2, 7, 25) (dual of [(74901, 7), 524073, 26]-NRT-code), using
- 24 times duplication [i] based on linear OOA(2230, 74901, F2, 7, 25) (dual of [(74901, 7), 524077, 26]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2230, 524307, F2, 25) (dual of [524307, 524077, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2230, 524308, F2, 25) (dual of [524308, 524078, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(2229, 524288, F2, 25) (dual of [524288, 524059, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2210, 524288, F2, 23) (dual of [524288, 524078, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(21, 20, F2, 1) (dual of [20, 19, 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(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2230, 524308, F2, 25) (dual of [524308, 524078, 26]-code), using
- OOA 7-folding [i] based on linear OA(2230, 524307, F2, 25) (dual of [524307, 524077, 26]-code), using
- 24 times duplication [i] based on linear OOA(2230, 74901, F2, 7, 25) (dual of [(74901, 7), 524077, 26]-NRT-code), using
(234−25, 234, 3701335)-Net in Base 2 — Upper bound on s
There is no (209, 234, 3701336)-net in base 2, because
- 1 times m-reduction [i] would yield (209, 233, 3701336)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 13803 518811 796747 151677 449234 963711 751205 073142 428018 205522 345981 300422 > 2233 [i]