Ihara constant over 𝔽_2

Description of constant

$C_{33}=A(2)$ is the Ihara constant over ${\mathbb{F}}_2$. <a href="#DM2013-def-Aq">[DM2013-def-Aq]</a>

For each integer $g\ge 1$, let

$$N_2(g):=\max \{\#X({\mathbb{F}}_2):X/{\mathbb{F}}_2\text{ a smooth projective geometrically integral curve of genus }g\}.$$

<a href="#DM2013-def-Nqg">[DM2013-def-Nqg]</a>

Then $$A(2):=\underset{g\to \infty }{\mathrm{limsup}}\frac{N_2(g)}{g}.$$

<a href="#DM2013-def-Aq">[DM2013-def-Aq]</a>

Known upper bounds

Known lower bounds

Additional comments and links

• Status. The exact value of $A(2)$ is unknown. As of the survey [Bee2022](#Bee2022), one has $$0.316999\dots \le A(2)\le \sqrt{2}-1\approx \mathrm{0.41421.}$$ <a href="#Bee2022-nonsquare-open">[Bee2022-nonsquare-open]</a> <a href="#DM2013-thm1.1">[DM2013-thm1.1]</a> <a href="#DM2013-dv-bound">[DM2013-dv-bound]</a>

• 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](#GS1995). For non-square $q$ (in particular $q=2$), no exact value is known. <a href="#DM2013-square-q">[DM2013-square-q]</a> <a href="#Bee2022-gs-tower">[Bee2022-gs-tower]</a> <a href="#Bee2022-nonsquare-open">[Bee2022-nonsquare-open]</a>

• 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](#TVZ1982), [Sti2009](#Sti2009). <a href="#Sti2005-TVZ">[Sti2005-TVZ]</a> <a href="#Bee2022-coding-theory">[Bee2022-coding-theory]</a>

• Tables for fixed genus. For small genera, records and exact maxima for $N_2(g)$ are tabulated at manypoints.org. <a href="#Fab2022-manypoints">[Fab2022-manypoints]</a>

• Surveys and background: [Bee2022](#Bee2022), [Sti2009](#Sti2009).

References

• <a id="Bee2022"></a>[Bee2022] Beelen, Peter. A survey on recursive towers and Ihara's constant. Preprint (2022). Google Scholar. arXiv PDF.
• <a id="Bee2022-lb-2-9"></a>[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].”
• <a id="Bee2022-lb-81-317"></a>[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: $A(2)\ge 2/9\approx 0.222222\dots$ [57, 55] $A(3)\ge 62/163\approx 0.380368\dots$ [53] $A(2)\ge 81/317\approx 0.255520\dots$ [53] $A(3)\ge 8/17\approx 0.470588\dots$ [59, 3]”
• <a id="Bee2022-lb-best"></a>[Bee2022-lb-best] loc: arXiv v1 PDF p.3, Section 2.1 (Bounds using class field theory). quote: “The following table gives an overview: $A(2)\ge 39/129\approx 0.302325\dots$ [45] $A(3)\ge 0.492876\dots$ [25] $A(2)\ge 0.316999\dots$ [25] The lower bounds for $A(2)$ and $A(3)$ found in [25] are currently the best known.”
• <a id="Bee2022-gs-tower"></a>[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.”
• <a id="Bee2022-nonsquare-open"></a>[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.”
• <a id="Bee2022-coding-theory"></a>[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].”

• <a id="DM2013"></a>[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
• <a id="DM2013-weil-bound"></a>[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 $q$.”
• <a id="DM2013-def-Nqg"></a>[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 $g$.”
• <a id="DM2013-def-Aq"></a>[DM2013-def-Aq] loc: arXiv v4 PDF p.1, Section 1 (Introduction). quote: “The Ihara constant is defined by $A(q):={\mathrm{limsup}}_{g\to \infty }{N}_q(g)/g$.”
• <a id="DM2013-dv-bound"></a>[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]).”
• <a id="DM2013-square-q"></a>[DM2013-square-q] loc: arXiv v4 PDF p.1, Section 1 (Introduction). quote: “and if $q$ is a square we have (see [12, 28]) $A(q)=\sqrt{q}-1$.”
• <a id="DM2013-prop1.1"></a>[DM2013-prop1.1] loc: arXiv v4 PDF p.2, Section 1 (Introduction). quote: “Among these results, the best lower bounds are $A(2)\ge 97/376=0.257979\dots$ by Xing and Yeo [31], and $A(3)\ge 12/25=0.48$ by Atiken and Hajir [8].”
• <a id="DM2013-prop1.2"></a>[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 $A(2)\ge 39/129=0.302325\dots$.”
• <a id="DM2013-thm1.1"></a>[DM2013-thm1.1] loc: arXiv v4 PDF p.2, Section 1 (Introduction). quote: “Theorem 1.3. $A(2)\ge 0.316999\dots$.”

• <a id="DV1983"></a>[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.

• <a id="Fab2022"></a>[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
• <a id="Fab2022-manypoints"></a>[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.”

• <a id="GS1995"></a>[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.

• <a id="NX1998"></a>[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.

• <a id="Sch1992"></a>[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)90079-590079-5). Google Scholar. Author PDF

• <a id="Ser1983"></a>[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

• <a id="Sti2005"></a>[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
• <a id="Sti2005-TVZ"></a>[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 $0\le \delta \le 1$.”

• <a id="Sti2009"></a>[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.

• <a id="TVZ1982"></a>[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.

• <a id="XY2007"></a>[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.

What counts as progress

• A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
• A formal proof (Lean) of a known bound, or a precise error in a claimed one.
• New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 87b4ebbd426f, Apache License 2.0; reformatted for this page. New records should also be reported there.