Best Known (165, 165+16, s)-Nets in Base 2
(165, 165+16, 524291)-Net over F2 — Constructive and digital
Digital (165, 181, 524291)-net over F2, using
- 22 times duplication [i] based on digital (163, 179, 524291)-net over F2, using
- t-expansion [i] based on digital (162, 179, 524291)-net over F2, using
- net defined by OOA [i] based on linear OOA(2179, 524291, F2, 17, 17) (dual of [(524291, 17), 8912768, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2179, 4194329, F2, 17) (dual of [4194329, 4194150, 18]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2178, 4194328, F2, 17) (dual of [4194328, 4194150, 18]-code), using
- construction X4 applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(2177, 4194304, F2, 17) (dual of [4194304, 4194127, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2155, 4194304, F2, 15) (dual of [4194304, 4194149, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(223, 24, F2, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,2)), using
- dual of repetition code with length 24 [i]
- linear OA(21, 24, F2, 1) (dual of [24, 23, 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 X4 applied to Ce(16) ⊂ Ce(14) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2178, 4194328, F2, 17) (dual of [4194328, 4194150, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2179, 4194329, F2, 17) (dual of [4194329, 4194150, 18]-code), using
- net defined by OOA [i] based on linear OOA(2179, 524291, F2, 17, 17) (dual of [(524291, 17), 8912768, 18]-NRT-code), using
- t-expansion [i] based on digital (162, 179, 524291)-net over F2, using
(165, 165+16, 838866)-Net over F2 — Digital
Digital (165, 181, 838866)-net over F2, using
- 21 times duplication [i] based on digital (164, 180, 838866)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2180, 838866, F2, 5, 16) (dual of [(838866, 5), 4194150, 17]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2180, 4194330, F2, 16) (dual of [4194330, 4194150, 17]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2177, 4194327, F2, 16) (dual of [4194327, 4194150, 17]-code), using
- 1 times truncation [i] based on linear OA(2178, 4194328, F2, 17) (dual of [4194328, 4194150, 18]-code), using
- construction X4 applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(2177, 4194304, F2, 17) (dual of [4194304, 4194127, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2155, 4194304, F2, 15) (dual of [4194304, 4194149, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(223, 24, F2, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,2)), using
- dual of repetition code with length 24 [i]
- linear OA(21, 24, F2, 1) (dual of [24, 23, 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 X4 applied to Ce(16) ⊂ Ce(14) [i] based on
- 1 times truncation [i] based on linear OA(2178, 4194328, F2, 17) (dual of [4194328, 4194150, 18]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2177, 4194327, F2, 16) (dual of [4194327, 4194150, 17]-code), using
- OOA 5-folding [i] based on linear OA(2180, 4194330, F2, 16) (dual of [4194330, 4194150, 17]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2180, 838866, F2, 5, 16) (dual of [(838866, 5), 4194150, 17]-NRT-code), using
(165, 165+16, large)-Net in Base 2 — Upper bound on s
There is no (165, 181, large)-net in base 2, because
- 14 times m-reduction [i] would yield (165, 167, large)-net in base 2, but