Best Known (121, 139, s)-Nets in Base 4
(121, 139, 116513)-Net over F4 — Constructive and digital
Digital (121, 139, 116513)-net over F4, using
- 41 times duplication [i] based on digital (120, 138, 116513)-net over F4, using
- net defined by OOA [i] based on linear OOA(4138, 116513, F4, 18, 18) (dual of [(116513, 18), 2097096, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(4138, 1048617, F4, 18) (dual of [1048617, 1048479, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4138, 1048619, F4, 18) (dual of [1048619, 1048481, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- linear OA(4131, 1048576, F4, 18) (dual of [1048576, 1048445, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(491, 1048576, F4, 13) (dual of [1048576, 1048485, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(4138, 1048619, F4, 18) (dual of [1048619, 1048481, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(4138, 1048617, F4, 18) (dual of [1048617, 1048479, 19]-code), using
- net defined by OOA [i] based on linear OOA(4138, 116513, F4, 18, 18) (dual of [(116513, 18), 2097096, 19]-NRT-code), using
(121, 139, 524312)-Net over F4 — Digital
Digital (121, 139, 524312)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4139, 524312, F4, 2, 18) (dual of [(524312, 2), 1048485, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4139, 1048624, F4, 18) (dual of [1048624, 1048485, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- linear OA(4131, 1048576, F4, 18) (dual of [1048576, 1048445, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(491, 1048576, F4, 13) (dual of [1048576, 1048485, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(48, 48, F4, 4) (dual of [48, 40, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(48, 85, F4, 4) (dual of [85, 77, 5]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- OOA 2-folding [i] based on linear OA(4139, 1048624, F4, 18) (dual of [1048624, 1048485, 19]-code), using
(121, 139, large)-Net in Base 4 — Upper bound on s
There is no (121, 139, large)-net in base 4, because
- 16 times m-reduction [i] would yield (121, 123, large)-net in base 4, but