Best Known (173−18, 173, s)-Nets in Base 3
(173−18, 173, 531444)-Net over F3 — Constructive and digital
Digital (155, 173, 531444)-net over F3, using
- t-expansion [i] based on digital (154, 173, 531444)-net over F3, using
- net defined by OOA [i] based on linear OOA(3173, 531444, F3, 19, 19) (dual of [(531444, 19), 10097263, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3173, 4782997, F3, 19) (dual of [4782997, 4782824, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3173, 4783001, F3, 19) (dual of [4783001, 4782828, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(3169, 4782969, F3, 19) (dual of [4782969, 4782800, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3141, 4782969, F3, 16) (dual of [4782969, 4782828, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(34, 32, F3, 2) (dual of [32, 28, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3173, 4783001, F3, 19) (dual of [4783001, 4782828, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3173, 4782997, F3, 19) (dual of [4782997, 4782824, 20]-code), using
- net defined by OOA [i] based on linear OOA(3173, 531444, F3, 19, 19) (dual of [(531444, 19), 10097263, 20]-NRT-code), using
(173−18, 173, 1594334)-Net over F3 — Digital
Digital (155, 173, 1594334)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3173, 1594334, F3, 3, 18) (dual of [(1594334, 3), 4782829, 19]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3170, 1594333, F3, 3, 18) (dual of [(1594333, 3), 4782829, 19]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3170, 4782999, F3, 18) (dual of [4782999, 4782829, 19]-code), using
- construction X4 applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(3169, 4782969, F3, 19) (dual of [4782969, 4782800, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3141, 4782969, F3, 16) (dual of [4782969, 4782828, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(329, 30, F3, 29) (dual of [30, 1, 30]-code or 30-arc in PG(28,3)), using
- dual of repetition code with length 30 [i]
- linear OA(31, 30, F3, 1) (dual of [30, 29, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(18) ⊂ Ce(15) [i] based on
- OOA 3-folding [i] based on linear OA(3170, 4782999, F3, 18) (dual of [4782999, 4782829, 19]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3170, 1594333, F3, 3, 18) (dual of [(1594333, 3), 4782829, 19]-NRT-code), using
(173−18, 173, large)-Net in Base 3 — Upper bound on s
There is no (155, 173, large)-net in base 3, because
- 16 times m-reduction [i] would yield (155, 157, large)-net in base 3, but