Best Known (45−9, 45, s)-Nets in Base 7
(45−9, 45, 29414)-Net over F7 — Constructive and digital
Digital (36, 45, 29414)-net over F7, using
- net defined by OOA [i] based on linear OOA(745, 29414, F7, 9, 9) (dual of [(29414, 9), 264681, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(745, 117657, F7, 9) (dual of [117657, 117612, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(745, 117658, F7, 9) (dual of [117658, 117613, 10]-code), using
- construction XX applied to Ce(8) ⊂ Ce(7) ⊂ Ce(5) [i] based on
- linear OA(743, 117649, F7, 9) (dual of [117649, 117606, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(737, 117649, F7, 8) (dual of [117649, 117612, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(731, 117649, F7, 6) (dual of [117649, 117618, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(70, 7, F7, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(71, 2, F7, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(8) ⊂ Ce(7) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(745, 117658, F7, 9) (dual of [117658, 117613, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(745, 117657, F7, 9) (dual of [117657, 117612, 10]-code), using
(45−9, 45, 115558)-Net over F7 — Digital
Digital (36, 45, 115558)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(745, 115558, F7, 9) (dual of [115558, 115513, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(745, 117658, F7, 9) (dual of [117658, 117613, 10]-code), using
- construction XX applied to Ce(8) ⊂ Ce(7) ⊂ Ce(5) [i] based on
- linear OA(743, 117649, F7, 9) (dual of [117649, 117606, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(737, 117649, F7, 8) (dual of [117649, 117612, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(731, 117649, F7, 6) (dual of [117649, 117618, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(70, 7, F7, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(71, 2, F7, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(8) ⊂ Ce(7) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(745, 117658, F7, 9) (dual of [117658, 117613, 10]-code), using
(45−9, 45, large)-Net in Base 7 — Upper bound on s
There is no (36, 45, large)-net in base 7, because
- 7 times m-reduction [i] would yield (36, 38, large)-net in base 7, but