Best Known (257−30, 257, s)-Nets in Base 4
(257−30, 257, 279624)-Net over F4 — Constructive and digital
Digital (227, 257, 279624)-net over F4, using
- 42 times duplication [i] based on digital (225, 255, 279624)-net over F4, using
- net defined by OOA [i] based on linear OOA(4255, 279624, F4, 30, 30) (dual of [(279624, 30), 8388465, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(4255, 4194360, F4, 30) (dual of [4194360, 4194105, 31]-code), using
- 4 times code embedding in larger space [i] based on linear OA(4251, 4194356, F4, 30) (dual of [4194356, 4194105, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(24) [i] based on
- linear OA(4243, 4194304, F4, 30) (dual of [4194304, 4194061, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4199, 4194304, F4, 25) (dual of [4194304, 4194105, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(48, 52, F4, 4) (dual of [52, 44, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(48, 85, F4, 4) (dual of [85, 77, 5]-code), using
- construction X applied to Ce(29) ⊂ Ce(24) [i] based on
- 4 times code embedding in larger space [i] based on linear OA(4251, 4194356, F4, 30) (dual of [4194356, 4194105, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(4255, 4194360, F4, 30) (dual of [4194360, 4194105, 31]-code), using
- net defined by OOA [i] based on linear OOA(4255, 279624, F4, 30, 30) (dual of [(279624, 30), 8388465, 31]-NRT-code), using
(257−30, 257, 1767675)-Net over F4 — Digital
Digital (227, 257, 1767675)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4257, 1767675, F4, 2, 30) (dual of [(1767675, 2), 3535093, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4257, 2097186, F4, 2, 30) (dual of [(2097186, 2), 4194115, 31]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4256, 2097186, F4, 2, 30) (dual of [(2097186, 2), 4194116, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4256, 4194372, F4, 30) (dual of [4194372, 4194116, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(22) [i] based on
- linear OA(4243, 4194304, F4, 30) (dual of [4194304, 4194061, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4188, 4194304, F4, 23) (dual of [4194304, 4194116, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(413, 68, F4, 6) (dual of [68, 55, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 70, F4, 6) (dual of [70, 57, 7]-code), using
- construction X applied to Ce(29) ⊂ Ce(22) [i] based on
- OOA 2-folding [i] based on linear OA(4256, 4194372, F4, 30) (dual of [4194372, 4194116, 31]-code), using
- 41 times duplication [i] based on linear OOA(4256, 2097186, F4, 2, 30) (dual of [(2097186, 2), 4194116, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4257, 2097186, F4, 2, 30) (dual of [(2097186, 2), 4194115, 31]-NRT-code), using
(257−30, 257, large)-Net in Base 4 — Upper bound on s
There is no (227, 257, large)-net in base 4, because
- 28 times m-reduction [i] would yield (227, 229, large)-net in base 4, but