Best Known (167, 167+31, s)-Nets in Base 2
(167, 167+31, 547)-Net over F2 — Constructive and digital
Digital (167, 198, 547)-net over F2, using
- 21 times duplication [i] based on digital (166, 197, 547)-net over F2, using
- net defined by OOA [i] based on linear OOA(2197, 547, F2, 31, 31) (dual of [(547, 31), 16760, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2197, 8206, F2, 31) (dual of [8206, 8009, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(2196, 8192, F2, 31) (dual of [8192, 7996, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2183, 8192, F2, 29) (dual of [8192, 8009, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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(30) ⊂ Ce(28) [i] based on
- OOA 15-folding and stacking with additional row [i] based on linear OA(2197, 8206, F2, 31) (dual of [8206, 8009, 32]-code), using
- net defined by OOA [i] based on linear OOA(2197, 547, F2, 31, 31) (dual of [(547, 31), 16760, 32]-NRT-code), using
(167, 167+31, 1822)-Net over F2 — Digital
Digital (167, 198, 1822)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2198, 1822, F2, 4, 31) (dual of [(1822, 4), 7090, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2198, 2051, F2, 4, 31) (dual of [(2051, 4), 8006, 32]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2197, 2051, F2, 4, 31) (dual of [(2051, 4), 8007, 32]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2197, 8204, F2, 31) (dual of [8204, 8007, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2197, 8206, F2, 31) (dual of [8206, 8009, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(2196, 8192, F2, 31) (dual of [8192, 7996, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2183, 8192, F2, 29) (dual of [8192, 8009, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(2197, 8206, F2, 31) (dual of [8206, 8009, 32]-code), using
- OOA 4-folding [i] based on linear OA(2197, 8204, F2, 31) (dual of [8204, 8007, 32]-code), using
- 21 times duplication [i] based on linear OOA(2197, 2051, F2, 4, 31) (dual of [(2051, 4), 8007, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2198, 2051, F2, 4, 31) (dual of [(2051, 4), 8006, 32]-NRT-code), using
(167, 167+31, 57693)-Net in Base 2 — Upper bound on s
There is no (167, 198, 57694)-net in base 2, because
- 1 times m-reduction [i] would yield (167, 197, 57694)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 200883 067354 215469 833309 742366 864714 743697 326537 964611 539468 > 2197 [i]