Best Known (154, 154+21, s)-Nets in Base 4
(154, 154+21, 419434)-Net over F4 — Constructive and digital
Digital (154, 175, 419434)-net over F4, using
- 44 times duplication [i] based on digital (150, 171, 419434)-net over F4, using
- net defined by OOA [i] based on linear OOA(4171, 419434, F4, 21, 21) (dual of [(419434, 21), 8807943, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4171, 4194341, F4, 21) (dual of [4194341, 4194170, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4171, 4194342, F4, 21) (dual of [4194342, 4194171, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(16) [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(4133, 4194304, F4, 17) (dual of [4194304, 4194171, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(45, 38, F4, 3) (dual of [38, 33, 4]-code or 38-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(4171, 4194342, F4, 21) (dual of [4194342, 4194171, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4171, 4194341, F4, 21) (dual of [4194341, 4194170, 22]-code), using
- net defined by OOA [i] based on linear OOA(4171, 419434, F4, 21, 21) (dual of [(419434, 21), 8807943, 22]-NRT-code), using
(154, 154+21, 1539821)-Net over F4 — Digital
Digital (154, 175, 1539821)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4175, 1539821, F4, 2, 21) (dual of [(1539821, 2), 3079467, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4175, 2097173, F4, 2, 21) (dual of [(2097173, 2), 4194171, 22]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(4171, 2097171, F4, 2, 21) (dual of [(2097171, 2), 4194171, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4171, 4194342, F4, 21) (dual of [4194342, 4194171, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(16) [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(4133, 4194304, F4, 17) (dual of [4194304, 4194171, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(45, 38, F4, 3) (dual of [38, 33, 4]-code or 38-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- OOA 2-folding [i] based on linear OA(4171, 4194342, F4, 21) (dual of [4194342, 4194171, 22]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(4171, 2097171, F4, 2, 21) (dual of [(2097171, 2), 4194171, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4175, 2097173, F4, 2, 21) (dual of [(2097173, 2), 4194171, 22]-NRT-code), using
(154, 154+21, large)-Net in Base 4 — Upper bound on s
There is no (154, 175, large)-net in base 4, because
- 19 times m-reduction [i] would yield (154, 156, large)-net in base 4, but