Best Known (248−34, 248, s)-Nets in Base 2
(248−34, 248, 1062)-Net over F2 — Constructive and digital
Digital (214, 248, 1062)-net over F2, using
- 22 times duplication [i] based on digital (212, 246, 1062)-net over F2, using
- trace code for nets [i] based on digital (7, 41, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 41, 177)-net over F64, using
(248−34, 248, 3938)-Net over F2 — Digital
Digital (214, 248, 3938)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2248, 3938, F2, 4, 34) (dual of [(3938, 4), 15504, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2248, 4105, F2, 4, 34) (dual of [(4105, 4), 16172, 35]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2247, 4105, F2, 4, 34) (dual of [(4105, 4), 16173, 35]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2247, 16420, F2, 34) (dual of [16420, 16173, 35]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2245, 16418, F2, 34) (dual of [16418, 16173, 35]-code), using
- 1 times truncation [i] based on linear OA(2246, 16419, F2, 35) (dual of [16419, 16173, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- linear OA(2239, 16384, F2, 35) (dual of [16384, 16145, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- 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(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(34) ⊂ Ce(30) [i] based on
- 1 times truncation [i] based on linear OA(2246, 16419, F2, 35) (dual of [16419, 16173, 36]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2245, 16418, F2, 34) (dual of [16418, 16173, 35]-code), using
- OOA 4-folding [i] based on linear OA(2247, 16420, F2, 34) (dual of [16420, 16173, 35]-code), using
- 21 times duplication [i] based on linear OOA(2247, 4105, F2, 4, 34) (dual of [(4105, 4), 16173, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2248, 4105, F2, 4, 34) (dual of [(4105, 4), 16172, 35]-NRT-code), using
(248−34, 248, 176758)-Net in Base 2 — Upper bound on s
There is no (214, 248, 176759)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 452 336542 041160 947297 569624 304422 002866 955980 139330 881002 720441 517970 123800 > 2248 [i]