Best Known (206, 206+28, s)-Nets in Base 4
(206, 206+28, 299594)-Net over F4 — Constructive and digital
Digital (206, 234, 299594)-net over F4, using
- t-expansion [i] based on digital (205, 234, 299594)-net over F4, using
- net defined by OOA [i] based on linear OOA(4234, 299594, F4, 29, 29) (dual of [(299594, 29), 8687992, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4234, 4194317, F4, 29) (dual of [4194317, 4194083, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4233, 4194316, F4, 29) (dual of [4194316, 4194083, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- linear OA(4232, 4194304, F4, 29) (dual of [4194304, 4194072, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4221, 4194304, F4, 27) (dual of [4194304, 4194083, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(41, 12, F4, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4233, 4194316, F4, 29) (dual of [4194316, 4194083, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4234, 4194317, F4, 29) (dual of [4194317, 4194083, 30]-code), using
- net defined by OOA [i] based on linear OOA(4234, 299594, F4, 29, 29) (dual of [(299594, 29), 8687992, 30]-NRT-code), using
(206, 206+28, 1398109)-Net over F4 — Digital
Digital (206, 234, 1398109)-net over F4, using
- 41 times duplication [i] based on digital (205, 233, 1398109)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4233, 1398109, F4, 3, 28) (dual of [(1398109, 3), 4194094, 29]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4233, 4194327, F4, 28) (dual of [4194327, 4194094, 29]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- linear OA(4232, 4194304, F4, 29) (dual of [4194304, 4194072, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4210, 4194304, F4, 26) (dual of [4194304, 4194094, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(41, 23, F4, 1) (dual of [23, 22, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- OOA 3-folding [i] based on linear OA(4233, 4194327, F4, 28) (dual of [4194327, 4194094, 29]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4233, 1398109, F4, 3, 28) (dual of [(1398109, 3), 4194094, 29]-NRT-code), using
(206, 206+28, large)-Net in Base 4 — Upper bound on s
There is no (206, 234, large)-net in base 4, because
- 26 times m-reduction [i] would yield (206, 208, large)-net in base 4, but