Ihara constant over $\mathbf{F}_2$

Description of constant

$C_{33}=A(2)$ is the Ihara constant over $\mathbb{F}_2$. [DM2013-def-Aq]

For each integer $g\ge 1$, let

[DM2013-def-Nqg]

Then \(A(2) := \limsup_{g\to\infty}\frac{N_{2}(g)}{g}.\)

[DM2013-def-Aq]

Known upper bounds

Known lower bounds

Additional comments and links

• Status. The exact value of $A(2)$ is unknown. As of the survey Bee2022, one has \(0.316999\ldots \ \le\ A(2)\ \le\ \sqrt2-1\approx 0.41421.\) [Bee2022-nonsquare-open] [DM2013-thm1.1] [DM2013-dv-bound]

• Square vs. non-square fields. For $q$ a square, the Drinfeld–Vlăduţ upper bound is sharp: $A(q)=\sqrt q-1$, via explicit towers of function fields (e.g. the Garcia–Stichtenoth tower) GS1995. For non-square $q$ (in particular $q=2$), no exact value is known. [DM2013-square-q] [Bee2022-gs-tower] [Bee2022-nonsquare-open]

• Connection to coding theory (TVZ bound). The quantity $A(q)$ controls the asymptotic performance of algebraic-geometry codes. In particular, the Tsfasman–Vlăduţ–Zink bound expresses an asymptotic rate–distance tradeoff in terms of $A(q)$; see TVZ1982, Sti2009. [Sti2005-TVZ] [Bee2022-coding-theory]

• Tables for fixed genus. For small genera, records and exact maxima for $N_2(g)$ are tabulated at manypoints.org. [Fab2022-manypoints]

• Surveys and background: Bee2022, Sti2009.

References

• [Bee2022] Beelen, Peter. A survey on recursive towers and Ihara’s constant. Preprint (2022). Google Scholar. arXiv PDF.
  • [Bee2022-lb-2-9] loc: arXiv v1 PDF p.3, Section 2.1 (Bounds using class field theory). quote: “Serre already demonstrated in his Harvard lectures, that for $q = 2$, class field theory can be used to show that $A(2) \ge 2/9$ [57, Theorem 5.11.1], a result that was also obtained using a different construction in [55].”
  • [Bee2022-lb-81-317] loc: arXiv v1 PDF p.3, Section 2.1 (Bounds using class field theory). quote: “The following table gives an overview: $A(2) \ge 2/9 \approx 0.222222\ldots$ [57, 55] $A(3) \ge 62/163 \approx 0.380368\ldots$ [53] $A(2) \ge 81/317 \approx 0.255520\ldots$ [53] $A(3) \ge 8/17 \approx 0.470588\ldots$ [59, 3]”
  • [Bee2022-lb-best] loc: arXiv v1 PDF p.3, Section 2.1 (Bounds using class field theory). quote: “The following table gives an overview: $A(2) \ge 39/129 \approx 0.302325\ldots$ [45] $A(3) \ge 0.492876\ldots$ [25] $A(2) \ge 0.316999\ldots$ [25] The lower bounds for $A(2)$ and $A(3)$ found in [25] are currently the best known.”
  • [Bee2022-gs-tower] loc: arXiv v1 PDF p.6, Section 3.1 (The first two Garcia–Stichtenoth towers). quote: “As $n \to \infty$ the ratio of number of places of degree one of $E_n$, denoted by $N_1(E_n)$, and the genus of $E_n$, denoted by $g(E_n)$, tends to $q - 1$, achieving the Drinfeld–Vladut bound.”
  • [Bee2022-nonsquare-open] loc: arXiv v1 PDF p.19, Section 5 (Recursive towers of function fields: non-square finite fields). quote: “The case of non-square finite fields is currently still open.”
  • [Bee2022-coding-theory] loc: arXiv v1 PDF p.4, Section 2.2 (Explicit equations for modular curves). quote: “One of the driving motivations for studying families of curves with many $\mathbb{F}_q$-rational points, is that using Goppa’s construction of error-correcting codes, such families can be used to find good families of such codes [62].”
• [DM2013] Duursma, Iwan; Mak, Koon-Ho. On lower bounds for the Ihara constants $A(2)$ and $A(3)$. Compositio Mathematica 149 (2013), 1108–1128. DOI: 10.1112/S0010437X12000796. Google Scholar. arXiv PDF
  • [DM2013-weil-bound] loc: arXiv v4 PDF p.1, Section 1 (Introduction). quote: “It is well-known that the Weil bound $\#X(\mathbb{F}_q) \le q + 1 + 2g\sqrt{q}$ is not sharp if $g$ is large compared to $q$.”
  • [DM2013-def-Nqg] loc: arXiv v4 PDF p.1, Section 1 (Introduction). quote: “Put $N_q(g) := \max \#X(\mathbb{F}_q)$, where the maximum is taken over all curves $X/\mathbb{F}_q$ with genus $g$.”
  • [DM2013-def-Aq] loc: arXiv v4 PDF p.1, Section 1 (Introduction). quote: “The Ihara constant is defined by $A(q) := \limsup_{g\to\infty} N_q(g)/g$.”
  • [DM2013-dv-bound] loc: arXiv v4 PDF p.1, Section 1 (Introduction). quote: “For any $q$, we have $A(q) \le \sqrt{q} - 1$ (see [4]).”
  • [DM2013-square-q] loc: arXiv v4 PDF p.1, Section 1 (Introduction). quote: “and if $q$ is a square we have (see [12, 28]) $A(q) = \sqrt{q} - 1$.”
  • [DM2013-prop1.1] loc: arXiv v4 PDF p.2, Section 1 (Introduction). quote: “Among these results, the best lower bounds are $A(2) \ge 97/376 = 0.257979\ldots$ by Xing and Yeo [31], and $A(3) \ge 12/25 = 0.48$ by Atiken and Hajir [8].”
  • [DM2013-prop1.2] loc: arXiv v4 PDF p.2, Section 1 (Introduction). quote: “In [13], Kuhnt obtained a better lower bound for $A(2)$, which says $A(2) \ge 39/129 = 0.302325\ldots$.”
  • [DM2013-thm1.1] loc: arXiv v4 PDF p.2, Section 1 (Introduction). quote: “Theorem 1.3. $A(2) \ge 0.316999\ldots$.”

• [DV1983] Drinfeld, V. G.; Vladut, S. G. Number of points of an algebraic curve. Functional Analysis and Its Applications 17 (1983), no. 1, 53–54. DOI: 10.1007/BF01083182. Google Scholar.

• [Fab2022] Faber, Xander; Grantham, Jon. Binary curves of small fixed genus and gonality with many rational points. Journal of Algebra 597 (2022), 24–46. DOI: 10.1016/j.jalgebra.2022.01.008. Google Scholar. arXiv PDF
  • [Fab2022-manypoints] loc: arXiv v2 PDF p.1, Section 1 (Introduction). quote: “Write $N_2(g)$ for the maximum number of rational points on a curve of genus $g$ over $\mathbb{F}_2$. The values in Table 1 come from manypoints.org.”

• [GS1995] Garcia, Arnaldo; Stichtenoth, Henning. A tower of Artin–Schreier extensions of function fields attaining the Drinfeld–Vladut bound. Inventiones Mathematicae 121 (1995), 211–222. DOI: 10.1007/BF01884295. Google Scholar.

• [NX1998] Niederreiter, Harald; Xing, Chaoping. Towers of global function fields with asymptotically many rational places and an improvement of the Gilbert–Varshamov bound. Mathematische Nachrichten 195 (1998), 171–186. DOI: 10.1002/mana.19981950110. Google Scholar.

• [Sch1992] Schoof, Rene. Algebraic curves over $\mathbb{F}_2$ with many rational points. Journal of Number Theory 41 (1992), no. 1, 6–14. DOI: 10.1016/0022-314X(92)90079-5. Google Scholar. Author PDF

• [Ser1983] Serre, Jean-Pierre. Sur le nombre des points rationnels d’une courbe algebrique sur un corps fini. C. R. Acad. Sci. Paris Ser. I Math. 296 (1983), 397–402. Google Scholar. Publisher entry

• [Sti2005] Stichtenoth, Henning. Transitive and self-dual codes attaining the Tsfasman-Vladut-Zink bound. IEEE Transactions on Information Theory 52 (5) (2006), 2218–2224. DOI: 10.1109/TIT.2006.872986. Google Scholar. arXiv PDF
  • [Sti2005-TVZ] loc: arXiv v1 PDF p.2, Section 1 (Introduction and Main Results). quote: “Then $\alpha_q(\delta) \ge 1 - \delta - A(q)^{-1}$ for $0 \le \delta \le 1$.”

• [Sti2009] Stichtenoth, Henning. Algebraic Function Fields and Codes. 2nd ed., Graduate Texts in Mathematics 254, Springer (2009). DOI: 10.1007/978-3-540-76878-4. Google Scholar.

• [TVZ1982] Tsfasman, M. A.; Vladut, S. G.; Zink, T. Modular curves, Shimura curves, and Goppa codes, better than the Varshamov–Gilbert bound. Mathematische Nachrichten 109 (1982), 21–28. DOI: 10.1002/mana.19821090103. Google Scholar.

• [XY2007] Xing, Chaoping; Yeo, Sze Ling. Algebraic curves with many points over the binary field. Journal of Algebra 311 (2007), no. 2, 775–780. DOI: 10.1016/j.jalgebra.2006.12.029. Google Scholar.

Contribution notes

Prepared with assistance from ChatGPT 5.2 Pro.