Best Known (87, 108, s)-Nets in Base 4
(87, 108, 1639)-Net over F4 — Constructive and digital
Digital (87, 108, 1639)-net over F4, using
- 41 times duplication [i] based on digital (86, 107, 1639)-net over F4, using
- net defined by OOA [i] based on linear OOA(4107, 1639, F4, 21, 21) (dual of [(1639, 21), 34312, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4107, 16391, F4, 21) (dual of [16391, 16284, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4107, 16392, F4, 21) (dual of [16392, 16285, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(4106, 16384, F4, 21) (dual of [16384, 16278, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(499, 16384, F4, 19) (dual of [16384, 16285, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(41, 8, F4, 1) (dual of [8, 7, 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
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(4107, 16392, F4, 21) (dual of [16392, 16285, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4107, 16391, F4, 21) (dual of [16391, 16284, 22]-code), using
- net defined by OOA [i] based on linear OOA(4107, 1639, F4, 21, 21) (dual of [(1639, 21), 34312, 22]-NRT-code), using
(87, 108, 8197)-Net over F4 — Digital
Digital (87, 108, 8197)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4108, 8197, F4, 2, 21) (dual of [(8197, 2), 16286, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4108, 16394, F4, 21) (dual of [16394, 16286, 22]-code), using
- construction XX applied to Ce(20) ⊂ Ce(18) ⊂ Ce(17) [i] based on
- linear OA(4106, 16384, F4, 21) (dual of [16384, 16278, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(499, 16384, F4, 19) (dual of [16384, 16285, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(492, 16384, F4, 18) (dual of [16384, 16292, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(41, 9, F4, 1) (dual of [9, 8, 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, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(20) ⊂ Ce(18) ⊂ Ce(17) [i] based on
- OOA 2-folding [i] based on linear OA(4108, 16394, F4, 21) (dual of [16394, 16286, 22]-code), using
(87, 108, 4177304)-Net in Base 4 — Upper bound on s
There is no (87, 108, 4177305)-net in base 4, because
- 1 times m-reduction [i] would yield (87, 107, 4177305)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 26328 099854 313544 475437 957676 446966 689441 509669 675355 192841 863256 > 4107 [i]