Best Known (108, 108+15, s)-Nets in Base 4
(108, 108+15, 599189)-Net over F4 — Constructive and digital
Digital (108, 123, 599189)-net over F4, using
- net defined by OOA [i] based on linear OOA(4123, 599189, F4, 15, 15) (dual of [(599189, 15), 8987712, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4123, 4194324, F4, 15) (dual of [4194324, 4194201, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(4123, 4194327, F4, 15) (dual of [4194327, 4194204, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(4122, 4194304, F4, 15) (dual of [4194304, 4194182, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(4100, 4194304, F4, 13) (dual of [4194304, 4194204, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(14) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(4123, 4194327, F4, 15) (dual of [4194327, 4194204, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4123, 4194324, F4, 15) (dual of [4194324, 4194201, 16]-code), using
(108, 108+15, 2074961)-Net over F4 — Digital
Digital (108, 123, 2074961)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4123, 2074961, F4, 2, 15) (dual of [(2074961, 2), 4149799, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4123, 2097164, F4, 2, 15) (dual of [(2097164, 2), 4194205, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4123, 4194328, F4, 15) (dual of [4194328, 4194205, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(4122, 4194304, F4, 15) (dual of [4194304, 4194182, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(4100, 4194304, F4, 13) (dual of [4194304, 4194204, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(423, 24, F4, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,4)), using
- dual of repetition code with length 24 [i]
- linear OA(41, 24, F4, 1) (dual of [24, 23, 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 X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- OOA 2-folding [i] based on linear OA(4123, 4194328, F4, 15) (dual of [4194328, 4194205, 16]-code), using
- discarding factors / shortening the dual code based on linear OOA(4123, 2097164, F4, 2, 15) (dual of [(2097164, 2), 4194205, 16]-NRT-code), using
(108, 108+15, large)-Net in Base 4 — Upper bound on s
There is no (108, 123, large)-net in base 4, because
- 13 times m-reduction [i] would yield (108, 110, large)-net in base 4, but