Best Known (149, 169, s)-Nets in Base 4
(149, 169, 419433)-Net over F4 — Constructive and digital
Digital (149, 169, 419433)-net over F4, using
- net defined by OOA [i] based on linear OOA(4169, 419433, F4, 20, 20) (dual of [(419433, 20), 8388491, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(4169, 4194330, F4, 20) (dual of [4194330, 4194161, 21]-code), using
- 2 times code embedding in larger space [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
- 2 times code embedding in larger space [i] based on linear OA(4167, 4194328, F4, 20) (dual of [4194328, 4194161, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(4169, 4194330, F4, 20) (dual of [4194330, 4194161, 21]-code), using
(149, 169, 1964152)-Net over F4 — Digital
Digital (149, 169, 1964152)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4169, 1964152, F4, 2, 20) (dual of [(1964152, 2), 3928135, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4169, 2097165, F4, 2, 20) (dual of [(2097165, 2), 4194161, 21]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] 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
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4167, 2097164, F4, 2, 20) (dual of [(2097164, 2), 4194161, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4169, 2097165, F4, 2, 20) (dual of [(2097165, 2), 4194161, 21]-NRT-code), using
(149, 169, large)-Net in Base 4 — Upper bound on s
There is no (149, 169, large)-net in base 4, because
- 18 times m-reduction [i] would yield (149, 151, large)-net in base 4, but