Best Known (222−31, 222, s)-Nets in Base 2
(222−31, 222, 1094)-Net over F2 — Constructive and digital
Digital (191, 222, 1094)-net over F2, using
- 25 times duplication [i] based on digital (186, 217, 1094)-net over F2, using
- net defined by OOA [i] based on linear OOA(2217, 1094, F2, 31, 31) (dual of [(1094, 31), 33697, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2217, 16411, F2, 31) (dual of [16411, 16194, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2217, 16416, F2, 31) (dual of [16416, 16199, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- linear OA(2211, 16384, F2, 31) (dual of [16384, 16173, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2183, 16384, F2, 27) (dual of [16384, 16201, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(2217, 16416, F2, 31) (dual of [16416, 16199, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2217, 16411, F2, 31) (dual of [16411, 16194, 32]-code), using
- net defined by OOA [i] based on linear OOA(2217, 1094, F2, 31, 31) (dual of [(1094, 31), 33697, 32]-NRT-code), using
(222−31, 222, 3488)-Net over F2 — Digital
Digital (191, 222, 3488)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2222, 3488, F2, 4, 31) (dual of [(3488, 4), 13730, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2222, 4105, F2, 4, 31) (dual of [(4105, 4), 16198, 32]-NRT-code), using
- 23 times duplication [i] based on linear OOA(2219, 4105, F2, 4, 31) (dual of [(4105, 4), 16201, 32]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2219, 16420, F2, 31) (dual of [16420, 16201, 32]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2218, 16419, F2, 31) (dual of [16419, 16201, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- linear OA(2211, 16384, F2, 31) (dual of [16384, 16173, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2183, 16384, F2, 27) (dual of [16384, 16201, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(27, 35, F2, 3) (dual of [35, 28, 4]-code or 35-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2218, 16419, F2, 31) (dual of [16419, 16201, 32]-code), using
- OOA 4-folding [i] based on linear OA(2219, 16420, F2, 31) (dual of [16420, 16201, 32]-code), using
- 23 times duplication [i] based on linear OOA(2219, 4105, F2, 4, 31) (dual of [(4105, 4), 16201, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2222, 4105, F2, 4, 31) (dual of [(4105, 4), 16198, 32]-NRT-code), using
(222−31, 222, 174939)-Net in Base 2 — Upper bound on s
There is no (191, 222, 174940)-net in base 2, because
- 1 times m-reduction [i] would yield (191, 221, 174940)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 3 370205 319339 013640 946542 742190 034302 644547 802962 609176 683206 113068 > 2221 [i]