Best Known (61, 84, s)-Nets in Base 9
(61, 84, 740)-Net over F9 — Constructive and digital
Digital (61, 84, 740)-net over F9, using
- 6 times m-reduction [i] based on digital (61, 90, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 45, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 45, 370)-net over F81, using
(61, 84, 6399)-Net over F9 — Digital
Digital (61, 84, 6399)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(984, 6399, F9, 23) (dual of [6399, 6315, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(984, 6574, F9, 23) (dual of [6574, 6490, 24]-code), using
- construction XX applied to C([0,11]) ⊂ C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(981, 6562, F9, 23) (dual of [6562, 6481, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 98−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(973, 6562, F9, 21) (dual of [6562, 6489, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 98−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(965, 6562, F9, 19) (dual of [6562, 6497, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 98−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(91, 10, F9, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- linear OA(91, 2, F9, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to C([0,11]) ⊂ C([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(984, 6574, F9, 23) (dual of [6574, 6490, 24]-code), using
(61, 84, large)-Net in Base 9 — Upper bound on s
There is no (61, 84, large)-net in base 9, because
- 21 times m-reduction [i] would yield (61, 63, large)-net in base 9, but