Best Known (123, 123+17, s)-Nets in Base 4
(123, 123+17, 524292)-Net over F4 — Constructive and digital
Digital (123, 140, 524292)-net over F4, using
- 42 times duplication [i] based on digital (121, 138, 524292)-net over F4, using
- net defined by OOA [i] based on linear OOA(4138, 524292, F4, 17, 17) (dual of [(524292, 17), 8912826, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4138, 4194337, F4, 17) (dual of [4194337, 4194199, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4138, 4194342, F4, 17) (dual of [4194342, 4194204, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- 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(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(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(16) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(4138, 4194342, F4, 17) (dual of [4194342, 4194204, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4138, 4194337, F4, 17) (dual of [4194337, 4194199, 18]-code), using
- net defined by OOA [i] based on linear OOA(4138, 524292, F4, 17, 17) (dual of [(524292, 17), 8912826, 18]-NRT-code), using
(123, 123+17, 1733505)-Net over F4 — Digital
Digital (123, 140, 1733505)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4140, 1733505, F4, 2, 17) (dual of [(1733505, 2), 3466870, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4140, 2097172, F4, 2, 17) (dual of [(2097172, 2), 4194204, 18]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4138, 2097171, F4, 2, 17) (dual of [(2097171, 2), 4194204, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4138, 4194342, F4, 17) (dual of [4194342, 4194204, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- 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(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(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(16) ⊂ Ce(12) [i] based on
- OOA 2-folding [i] based on linear OA(4138, 4194342, F4, 17) (dual of [4194342, 4194204, 18]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4138, 2097171, F4, 2, 17) (dual of [(2097171, 2), 4194204, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4140, 2097172, F4, 2, 17) (dual of [(2097172, 2), 4194204, 18]-NRT-code), using
(123, 123+17, large)-Net in Base 4 — Upper bound on s
There is no (123, 140, large)-net in base 4, because
- 15 times m-reduction [i] would yield (123, 125, large)-net in base 4, but