Best Known (223, 223+31, s)-Nets in Base 4
(223, 223+31, 279620)-Net over F4 — Constructive and digital
Digital (223, 254, 279620)-net over F4, using
- net defined by OOA [i] based on linear OOA(4254, 279620, F4, 31, 31) (dual of [(279620, 31), 8667966, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4254, 4194301, F4, 31) (dual of [4194301, 4194047, 32]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(4254, 4194304, F4, 31) (dual of [4194304, 4194050, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4254, 4194301, F4, 31) (dual of [4194301, 4194047, 32]-code), using
(223, 223+31, 1398105)-Net over F4 — Digital
Digital (223, 254, 1398105)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4254, 1398105, F4, 3, 31) (dual of [(1398105, 3), 4194061, 32]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4254, 4194315, F4, 31) (dual of [4194315, 4194061, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(29) [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(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(40, 11, F4, 0) (dual of [11, 11, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(30) ⊂ Ce(29) [i] based on
- OOA 3-folding [i] based on linear OA(4254, 4194315, F4, 31) (dual of [4194315, 4194061, 32]-code), using
(223, 223+31, large)-Net in Base 4 — Upper bound on s
There is no (223, 254, large)-net in base 4, because
- 29 times m-reduction [i] would yield (223, 225, large)-net in base 4, but