Best Known (54−13, 54, s)-Nets in Base 2
(54−13, 54, 84)-Net over F2 — Constructive and digital
Digital (41, 54, 84)-net over F2, using
- trace code for nets [i] based on digital (5, 18, 28)-net over F8, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 5 and N(F) ≥ 28, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
(54−13, 54, 138)-Net over F2 — Digital
Digital (41, 54, 138)-net over F2, using
- 21 times duplication [i] based on digital (40, 53, 138)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(253, 138, F2, 2, 13) (dual of [(138, 2), 223, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(253, 276, F2, 13) (dual of [276, 223, 14]-code), using
- 1 times code embedding in larger space [i] based on linear OA(252, 275, F2, 13) (dual of [275, 223, 14]-code), using
- adding a parity check bit [i] based on linear OA(251, 274, F2, 12) (dual of [274, 223, 13]-code), using
- construction XX applied to C1 = C([253,8]), C2 = C([1,10]), C3 = C1 + C2 = C([1,8]), and C∩ = C1 ∩ C2 = C([253,10]) [i] based on
- linear OA(241, 255, F2, 11) (dual of [255, 214, 12]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,8}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(240, 255, F2, 10) (dual of [255, 215, 11]-code), using the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(249, 255, F2, 13) (dual of [255, 206, 14]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,10}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(232, 255, F2, 8) (dual of [255, 223, 9]-code), using the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(21, 10, F2, 1) (dual of [10, 9, 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
- linear OA(21, 9, F2, 1) (dual of [9, 8, 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 (see above)
- construction XX applied to C1 = C([253,8]), C2 = C([1,10]), C3 = C1 + C2 = C([1,8]), and C∩ = C1 ∩ C2 = C([253,10]) [i] based on
- adding a parity check bit [i] based on linear OA(251, 274, F2, 12) (dual of [274, 223, 13]-code), using
- 1 times code embedding in larger space [i] based on linear OA(252, 275, F2, 13) (dual of [275, 223, 14]-code), using
- OOA 2-folding [i] based on linear OA(253, 276, F2, 13) (dual of [276, 223, 14]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(253, 138, F2, 2, 13) (dual of [(138, 2), 223, 14]-NRT-code), using
(54−13, 54, 1357)-Net in Base 2 — Upper bound on s
There is no (41, 54, 1358)-net in base 2, because
- 1 times m-reduction [i] would yield (41, 53, 1358)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 9042 354763 474012 > 253 [i]