Best Known (45, 45+13, s)-Nets in Base 7
(45, 45+13, 2803)-Net over F7 — Constructive and digital
Digital (45, 58, 2803)-net over F7, using
- net defined by OOA [i] based on linear OOA(758, 2803, F7, 13, 13) (dual of [(2803, 13), 36381, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(758, 16819, F7, 13) (dual of [16819, 16761, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(758, 16820, F7, 13) (dual of [16820, 16762, 14]-code), using
- construction XX applied to Ce(12) ⊂ Ce(10) ⊂ Ce(9) [i] based on
- linear OA(756, 16807, F7, 13) (dual of [16807, 16751, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(746, 16807, F7, 11) (dual of [16807, 16761, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(741, 16807, F7, 10) (dual of [16807, 16766, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(71, 12, F7, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(12) ⊂ Ce(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(758, 16820, F7, 13) (dual of [16820, 16762, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(758, 16819, F7, 13) (dual of [16819, 16761, 14]-code), using
(45, 45+13, 16820)-Net over F7 — Digital
Digital (45, 58, 16820)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(758, 16820, F7, 13) (dual of [16820, 16762, 14]-code), using
- construction XX applied to Ce(12) ⊂ Ce(10) ⊂ Ce(9) [i] based on
- linear OA(756, 16807, F7, 13) (dual of [16807, 16751, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(746, 16807, F7, 11) (dual of [16807, 16761, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(741, 16807, F7, 10) (dual of [16807, 16766, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(71, 12, F7, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(12) ⊂ Ce(10) ⊂ Ce(9) [i] based on
(45, 45+13, large)-Net in Base 7 — Upper bound on s
There is no (45, 58, large)-net in base 7, because
- 11 times m-reduction [i] would yield (45, 47, large)-net in base 7, but