Best Known (42, 42+10, s)-Nets in Base 4
(42, 42+10, 3278)-Net over F4 — Constructive and digital
Digital (42, 52, 3278)-net over F4, using
- 42 times duplication [i] based on digital (40, 50, 3278)-net over F4, using
- net defined by OOA [i] based on linear OOA(450, 3278, F4, 10, 10) (dual of [(3278, 10), 32730, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(450, 16390, F4, 10) (dual of [16390, 16340, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(450, 16391, F4, 10) (dual of [16391, 16341, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(450, 16384, F4, 10) (dual of [16384, 16334, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(443, 16384, F4, 9) (dual of [16384, 16341, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(40, 7, F4, 0) (dual of [7, 7, 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 X applied to Ce(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(450, 16391, F4, 10) (dual of [16391, 16341, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(450, 16390, F4, 10) (dual of [16390, 16340, 11]-code), using
- net defined by OOA [i] based on linear OOA(450, 3278, F4, 10, 10) (dual of [(3278, 10), 32730, 11]-NRT-code), using
(42, 42+10, 8638)-Net over F4 — Digital
Digital (42, 52, 8638)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(452, 8638, F4, 10) (dual of [8638, 8586, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(452, 16394, F4, 10) (dual of [16394, 16342, 11]-code), using
- construction XX applied to Ce(9) ⊂ Ce(8) ⊂ Ce(6) [i] based on
- linear OA(450, 16384, F4, 10) (dual of [16384, 16334, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(443, 16384, F4, 9) (dual of [16384, 16341, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(436, 16384, F4, 7) (dual of [16384, 16348, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(40, 8, F4, 0) (dual of [8, 8, 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
- linear OA(41, 2, F4, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(9) ⊂ Ce(8) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(452, 16394, F4, 10) (dual of [16394, 16342, 11]-code), using
(42, 42+10, 1585396)-Net in Base 4 — Upper bound on s
There is no (42, 52, 1585397)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 20 282440 468588 104333 942670 124596 > 452 [i]