Best Known (42−5, 42, s)-Nets in Base 3
(42−5, 42, 797171)-Net over F3 — Constructive and digital
Digital (37, 42, 797171)-net over F3, using
- net defined by OOA [i] based on linear OOA(342, 797171, F3, 5, 5) (dual of [(797171, 5), 3985813, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(342, 797172, F3, 2, 5) (dual of [(797172, 2), 1594302, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(32, 4, F3, 2, 2) (dual of [(4, 2), 6, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(2;6,3) [i]
- linear OOA(340, 797168, F3, 2, 5) (dual of [(797168, 2), 1594296, 6]-NRT-code), using
- OOA 2-folding [i] based on linear OA(340, 1594336, F3, 5) (dual of [1594336, 1594296, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(340, 1594323, F3, 5) (dual of [1594323, 1594283, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(327, 1594323, F3, 4) (dual of [1594323, 1594296, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(30, 13, F3, 0) (dual of [13, 13, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- OOA 2-folding [i] based on linear OA(340, 1594336, F3, 5) (dual of [1594336, 1594296, 6]-code), using
- linear OOA(32, 4, F3, 2, 2) (dual of [(4, 2), 6, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(342, 797172, F3, 2, 5) (dual of [(797172, 2), 1594302, 6]-NRT-code), using
(42−5, 42, 1594340)-Net over F3 — Digital
Digital (37, 42, 1594340)-net over F3, using
- net defined by OOA [i] based on linear OOA(342, 1594340, F3, 5, 5) (dual of [(1594340, 5), 7971658, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(342, 1594340, F3, 4, 5) (dual of [(1594340, 4), 6377318, 6]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(342, 1594340, F3, 5) (dual of [1594340, 1594298, 6]-code), using
- (u, u+v)-construction [i] based on
- linear OA(32, 4, F3, 2) (dual of [4, 2, 3]-code or 4-arc in PG(1,3)), using
- extended Reed–Solomon code RSe(2,3) [i]
- Hamming code H(2,3) [i]
- Simplex code S(2,3) [i]
- the Tetracode [i]
- linear OA(340, 1594336, F3, 5) (dual of [1594336, 1594296, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(340, 1594323, F3, 5) (dual of [1594323, 1594283, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(327, 1594323, F3, 4) (dual of [1594323, 1594296, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(30, 13, F3, 0) (dual of [13, 13, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(32, 4, F3, 2) (dual of [4, 2, 3]-code or 4-arc in PG(1,3)), using
- (u, u+v)-construction [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(342, 1594340, F3, 5) (dual of [1594340, 1594298, 6]-code), using
- appending kth column [i] based on linear OOA(342, 1594340, F3, 4, 5) (dual of [(1594340, 4), 6377318, 6]-NRT-code), using
(42−5, 42, large)-Net in Base 3 — Upper bound on s
There is no (37, 42, large)-net in base 3, because
- 3 times m-reduction [i] would yield (37, 39, large)-net in base 3, but