Best Known (257−31, 257, s)-Nets in Base 4
(257−31, 257, 279621)-Net over F4 — Constructive and digital
Digital (226, 257, 279621)-net over F4, using
- 42 times duplication [i] based on digital (224, 255, 279621)-net over F4, using
- net defined by OOA [i] based on linear OOA(4255, 279621, F4, 31, 31) (dual of [(279621, 31), 8667996, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4255, 4194316, F4, 31) (dual of [4194316, 4194061, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4255, 4194327, F4, 31) (dual of [4194327, 4194072, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(4254, 4194304, F4, 31) (dual of [4194304, 4194050, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- 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(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(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(4255, 4194327, F4, 31) (dual of [4194327, 4194072, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4255, 4194316, F4, 31) (dual of [4194316, 4194061, 32]-code), using
- net defined by OOA [i] based on linear OOA(4255, 279621, F4, 31, 31) (dual of [(279621, 31), 8667996, 32]-NRT-code), using
(257−31, 257, 1398109)-Net over F4 — Digital
Digital (226, 257, 1398109)-net over F4, using
- 42 times duplication [i] based on digital (224, 255, 1398109)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4255, 1398109, F4, 3, 31) (dual of [(1398109, 3), 4194072, 32]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4255, 4194327, F4, 31) (dual of [4194327, 4194072, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(4254, 4194304, F4, 31) (dual of [4194304, 4194050, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- 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(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(30) ⊂ Ce(28) [i] based on
- OOA 3-folding [i] based on linear OA(4255, 4194327, F4, 31) (dual of [4194327, 4194072, 32]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4255, 1398109, F4, 3, 31) (dual of [(1398109, 3), 4194072, 32]-NRT-code), using
(257−31, 257, large)-Net in Base 4 — Upper bound on s
There is no (226, 257, large)-net in base 4, because
- 29 times m-reduction [i] would yield (226, 228, large)-net in base 4, but