Best Known (124, 124+17, s)-Nets in Base 4
(124, 124+17, 524293)-Net over F4 — Constructive and digital
Digital (124, 141, 524293)-net over F4, using
- net defined by OOA [i] based on linear OOA(4141, 524293, F4, 17, 17) (dual of [(524293, 17), 8912840, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4141, 4194345, F4, 17) (dual of [4194345, 4194204, 18]-code), using
- 3 times code embedding in larger space [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
- 3 times code embedding in larger space [i] 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(4141, 4194345, F4, 17) (dual of [4194345, 4194204, 18]-code), using
(124, 124+17, 1913946)-Net over F4 — Digital
Digital (124, 141, 1913946)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4141, 1913946, F4, 2, 17) (dual of [(1913946, 2), 3827751, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4141, 2097172, F4, 2, 17) (dual of [(2097172, 2), 4194203, 18]-NRT-code), using
- 41 times duplication [i] 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
- 41 times duplication [i] based on linear OOA(4140, 2097172, F4, 2, 17) (dual of [(2097172, 2), 4194204, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4141, 2097172, F4, 2, 17) (dual of [(2097172, 2), 4194203, 18]-NRT-code), using
(124, 124+17, large)-Net in Base 4 — Upper bound on s
There is no (124, 141, large)-net in base 4, because
- 15 times m-reduction [i] would yield (124, 126, large)-net in base 4, but