Best Known (67, 82, s)-Nets in Base 4
(67, 82, 2343)-Net over F4 — Constructive and digital
Digital (67, 82, 2343)-net over F4, using
- 41 times duplication [i] based on digital (66, 81, 2343)-net over F4, using
- net defined by OOA [i] based on linear OOA(481, 2343, F4, 15, 15) (dual of [(2343, 15), 35064, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(481, 16402, F4, 15) (dual of [16402, 16321, 16]-code), using
- 2 times code embedding in larger space [i] based on linear OA(479, 16400, F4, 15) (dual of [16400, 16321, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(478, 16384, F4, 15) (dual of [16384, 16306, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(415, 16, F4, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,4)), using
- dual of repetition code with length 16 [i]
- linear OA(41, 16, F4, 1) (dual of [16, 15, 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 X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(479, 16400, F4, 15) (dual of [16400, 16321, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(481, 16402, F4, 15) (dual of [16402, 16321, 16]-code), using
- net defined by OOA [i] based on linear OOA(481, 2343, F4, 15, 15) (dual of [(2343, 15), 35064, 16]-NRT-code), using
(67, 82, 10646)-Net over F4 — Digital
Digital (67, 82, 10646)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(482, 10646, F4, 15) (dual of [10646, 10564, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(482, 16404, F4, 15) (dual of [16404, 16322, 16]-code), using
- construction XX applied to Ce(14) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- linear OA(478, 16384, F4, 15) (dual of [16384, 16306, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(457, 16384, F4, 11) (dual of [16384, 16327, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(41, 17, F4, 1) (dual of [17, 16, 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(41, 3, F4, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- Reed–Solomon code RS(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- construction XX applied to Ce(14) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(482, 16404, F4, 15) (dual of [16404, 16322, 16]-code), using
(67, 82, large)-Net in Base 4 — Upper bound on s
There is no (67, 82, large)-net in base 4, because
- 13 times m-reduction [i] would yield (67, 69, large)-net in base 4, but