Best Known (212−115, 212, s)-Nets in Base 2
(212−115, 212, 54)-Net over F2 — Constructive and digital
Digital (97, 212, 54)-net over F2, using
- t-expansion [i] based on digital (95, 212, 54)-net over F2, using
- net from sequence [i] based on digital (95, 53)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, 1 place with degree 2, and 5 places with degree 6 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (95, 53)-sequence over F2, using
(212−115, 212, 65)-Net over F2 — Digital
Digital (97, 212, 65)-net over F2, using
- t-expansion [i] based on digital (95, 212, 65)-net over F2, using
- net from sequence [i] based on digital (95, 64)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 95 and N(F) ≥ 65, using
- net from sequence [i] based on digital (95, 64)-sequence over F2, using
(212−115, 212, 204)-Net over F2 — Upper bound on s (digital)
There is no digital (97, 212, 205)-net over F2, because
- 17 times m-reduction [i] would yield digital (97, 195, 205)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2195, 205, F2, 98) (dual of [205, 10, 99]-code), but
- residual code [i] would yield linear OA(297, 106, F2, 49) (dual of [106, 9, 50]-code), but
- 1 times truncation [i] would yield linear OA(296, 105, F2, 48) (dual of [105, 9, 49]-code), but
- residual code [i] would yield linear OA(248, 56, F2, 24) (dual of [56, 8, 25]-code), but
- adding a parity check bit [i] would yield linear OA(249, 57, F2, 25) (dual of [57, 8, 26]-code), but
- “BJV†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(249, 57, F2, 25) (dual of [57, 8, 26]-code), but
- residual code [i] would yield linear OA(248, 56, F2, 24) (dual of [56, 8, 25]-code), but
- 1 times truncation [i] would yield linear OA(296, 105, F2, 48) (dual of [105, 9, 49]-code), but
- residual code [i] would yield linear OA(297, 106, F2, 49) (dual of [106, 9, 50]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2195, 205, F2, 98) (dual of [205, 10, 99]-code), but
(212−115, 212, 206)-Net in Base 2 — Upper bound on s
There is no (97, 212, 207)-net in base 2, because
- 11 times m-reduction [i] would yield (97, 201, 207)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2201, 207, S2, 104), but
- adding a parity check bit [i] would yield OA(2202, 208, S2, 105), but
- the (dual) Plotkin bound shows that M ≥ 411 376139 330301 510538 742295 639337 626245 683966 408394 965837 152256 / 53 > 2202 [i]
- adding a parity check bit [i] would yield OA(2202, 208, S2, 105), but
- extracting embedded orthogonal array [i] would yield OA(2201, 207, S2, 104), but