Best Known (167−20, 167, s)-Nets in Base 4
(167−20, 167, 419432)-Net over F4 — Constructive and digital
Digital (147, 167, 419432)-net over F4, using
- net defined by OOA [i] based on linear OOA(4167, 419432, F4, 20, 20) (dual of [(419432, 20), 8388473, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(4167, 4194320, F4, 20) (dual of [4194320, 4194153, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4167, 4194327, F4, 20) (dual of [4194327, 4194160, 21]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- linear OA(4166, 4194304, F4, 21) (dual of [4194304, 4194138, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(4144, 4194304, F4, 18) (dual of [4194304, 4194160, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(20) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4167, 4194327, F4, 20) (dual of [4194327, 4194160, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(4167, 4194320, F4, 20) (dual of [4194320, 4194153, 21]-code), using
(167−20, 167, 1668568)-Net over F4 — Digital
Digital (147, 167, 1668568)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4167, 1668568, F4, 2, 20) (dual of [(1668568, 2), 3336969, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4167, 2097164, F4, 2, 20) (dual of [(2097164, 2), 4194161, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4167, 4194328, F4, 20) (dual of [4194328, 4194161, 21]-code), using
- construction X4 applied to Ce(20) ⊂ Ce(17) [i] based on
- linear OA(4166, 4194304, F4, 21) (dual of [4194304, 4194138, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(4144, 4194304, F4, 18) (dual of [4194304, 4194160, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(20) ⊂ Ce(17) [i] based on
- OOA 2-folding [i] based on linear OA(4167, 4194328, F4, 20) (dual of [4194328, 4194161, 21]-code), using
- discarding factors / shortening the dual code based on linear OOA(4167, 2097164, F4, 2, 20) (dual of [(2097164, 2), 4194161, 21]-NRT-code), using
(167−20, 167, large)-Net in Base 4 — Upper bound on s
There is no (147, 167, large)-net in base 4, because
- 18 times m-reduction [i] would yield (147, 149, large)-net in base 4, but