Best Known (200−27, 200, s)-Nets in Base 2
(200−27, 200, 2522)-Net over F2 — Constructive and digital
Digital (173, 200, 2522)-net over F2, using
- net defined by OOA [i] based on linear OOA(2200, 2522, F2, 27, 27) (dual of [(2522, 27), 67894, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2200, 32787, F2, 27) (dual of [32787, 32587, 28]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2197, 32784, F2, 27) (dual of [32784, 32587, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- linear OA(2196, 32768, F2, 27) (dual of [32768, 32572, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2181, 32768, F2, 25) (dual of [32768, 32587, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(21, 16, F2, 1) (dual of [16, 15, 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(26) ⊂ Ce(24) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(2197, 32784, F2, 27) (dual of [32784, 32587, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2200, 32787, F2, 27) (dual of [32787, 32587, 28]-code), using
(200−27, 200, 5464)-Net over F2 — Digital
Digital (173, 200, 5464)-net over F2, using
- 23 times duplication [i] based on digital (170, 197, 5464)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2197, 5464, F2, 6, 27) (dual of [(5464, 6), 32587, 28]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2197, 32784, F2, 27) (dual of [32784, 32587, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- linear OA(2196, 32768, F2, 27) (dual of [32768, 32572, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2181, 32768, F2, 25) (dual of [32768, 32587, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(21, 16, F2, 1) (dual of [16, 15, 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(26) ⊂ Ce(24) [i] based on
- OOA 6-folding [i] based on linear OA(2197, 32784, F2, 27) (dual of [32784, 32587, 28]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2197, 5464, F2, 6, 27) (dual of [(5464, 6), 32587, 28]-NRT-code), using
(200−27, 200, 229850)-Net in Base 2 — Upper bound on s
There is no (173, 200, 229851)-net in base 2, because
- 1 times m-reduction [i] would yield (173, 199, 229851)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 803506 717511 635149 501677 758224 359775 448239 198649 941521 689552 > 2199 [i]