Best Known (139−20, 139, s)-Nets in Base 4
(139−20, 139, 26216)-Net over F4 — Constructive and digital
Digital (119, 139, 26216)-net over F4, using
- t-expansion [i] based on digital (118, 139, 26216)-net over F4, using
- net defined by OOA [i] based on linear OOA(4139, 26216, F4, 21, 21) (dual of [(26216, 21), 550397, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4139, 262161, F4, 21) (dual of [262161, 262022, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4139, 262165, F4, 21) (dual of [262165, 262026, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- linear OA(4136, 262144, F4, 21) (dual of [262144, 262008, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(4118, 262144, F4, 18) (dual of [262144, 262026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4139, 262165, F4, 21) (dual of [262165, 262026, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4139, 262161, F4, 21) (dual of [262161, 262022, 22]-code), using
- net defined by OOA [i] based on linear OOA(4139, 26216, F4, 21, 21) (dual of [(26216, 21), 550397, 22]-NRT-code), using
(139−20, 139, 131083)-Net over F4 — Digital
Digital (119, 139, 131083)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4139, 131083, F4, 2, 20) (dual of [(131083, 2), 262027, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4139, 262166, F4, 20) (dual of [262166, 262027, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4139, 262167, F4, 20) (dual of [262167, 262028, 21]-code), using
- construction XX applied to Ce(20) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- linear OA(4136, 262144, F4, 21) (dual of [262144, 262008, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(4118, 262144, F4, 18) (dual of [262144, 262026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(4109, 262144, F4, 17) (dual of [262144, 262035, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(41, 21, F4, 1) (dual of [21, 20, 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
- linear OA(40, 2, F4, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction XX applied to Ce(20) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(4139, 262167, F4, 20) (dual of [262167, 262028, 21]-code), using
- OOA 2-folding [i] based on linear OA(4139, 262166, F4, 20) (dual of [262166, 262027, 21]-code), using
(139−20, 139, large)-Net in Base 4 — Upper bound on s
There is no (119, 139, large)-net in base 4, because
- 18 times m-reduction [i] would yield (119, 121, large)-net in base 4, but