On a conjecture of Goppa on codes and counting points hyperelliptic curves

What is the best code of a given length? This natural, but very hard, question motivates the following definition.

Definition: Let B_q(n,d) denote the largest k such that there exists a [n,k,d]-code in GF(q)^n.

Determining B_q(n,d) is one of the main problems in the theory of error-correcting codes.

Let C_i be a sequence of linear codes with parameters [n_i,k_i,d_i] such that n_i\rightarrow \infty as i \to \infty, and such that both the limits R=\lim_{i\to \infty}\frac{k_i}{n_i} and \delta=\lim_{i\rightarrow \infty}\frac{d_i}{n_i} exist. The Singleton (upper) bound implies R+\delta\leq 1.

Let \Sigma_q denote the set of all (\delta,R)\in [0,1]^2 such that there exists a sequence C_i, i=1,2,..., of [n_i,k_i,d_i]-codes for which \lim_{i\to \infty} d_i/n_1=\delta and \lim_{i\rightarrow \infty} k_i/n_i=R.

Theorem (Manin [SS]): There exists a continuous decreasing function \alpha_q:[0,1]\to [0,1], such that

  1. \alpha_q is strictly decreasing on [0,\frac{q-1}{q}],
  2. \alpha_q(0)=1,
  3. if {\frac{q-1}{q}}\leq x\leq 1 then \alpha_q(x)=0,
  4. \Sigma_q=\{(\delta,R)\in [0,1]^2\ |\ 0\leq R\leq \alpha_q(\delta)\}.

Theorem (Gilbert-Varshamov): We have \alpha_q(x)\geq 1- x\log_q(q-1)-x\log_q(x)-(1-x)\log_q(1-x). In other words, for each fixed \epsilon >0, there exists an [n,k,d]-code C (which may depend on \epsilon) with R(C)+\delta(C)\geq 1- \delta(C)\log_q({\frac{q-1}{q}})-\delta(C)\log_q(\delta(C))-(1-\delta(C))\log_q(1-\delta(C))-\epsilon. In particular, in the case q=2, \alpha_q(x)\geq1-H_2(\delta), where H_q(\delta)=\delta\cdot \log_q(q-1)-\delta\log_q(\delta)-(1-\delta)\log_q(1-\delta) is the entropy function (for a q-ary channel).

Plot of the Gilbert-Varshamov (dotted), Elias (red), Plotkin (dashed), Singleton (dash-dotted), Hamming (green), and MRRW (stepped) curves using SAGE:

asymptotic coding theory bounds
Not a single value of \alpha_q(x) is known for 0<x<{\frac{q-1}{q}}!

Goppa’s conjecture: \alpha_q(x) = 1-H_2(\delta). (In other words, the Gilbert-Varshamov lower bound is actually attained in the case q=2.)

Though my guess is that Goppa’s conjecture is true, it is known (thanks to work of Zink, Vladut and Varshamov) that the q-ary analog of Goppa’s conjecture is false for q>49. On the other hand, we have the following theorem [J].

Theorem: If B(1.77,p) is true for infinitely many primes p with p\equiv 1\pmod 4 then Goppa’s conjecture is false.

The rest of this post will be devoted to explaining the notation in this Theorem. The notation B(c,p) means: “For all subsets S\subset GF(p), |X_S(GF(p))| \leq  c\cdot p holds.” The notation X_S means: For each non-empty subset S\subset GF(p), define the hyperelliptic curve X_S by y^2=f_S(x), where f_S(x) = \prod_{a\in S}(x-a).

It may seem strange that a statement about the asymptotic bounds of error-correcting curves is related to the number of points on hyperelliptic curves, but for details see [J]. Unfortunately, it appears much more needs to be known about hyperelliptic curves over finite fields if one is to try to use this connection to prove or disprove Goppa’s conjecture.

References:
[J] D. Joyner, On quadratic residue codes and hyperelliptic curves. Discrete Mathematics & Theoretical Computer Science, Vol 10, No 1 (2008).
[SS] S. Shokranian and M.A. Shokrollahi, Coding theory and bilinear complexity, Scientific Series of the International Bureau, KFA Julich Vol. 21, 1994.


Here’s how the above plot is constructed using SAGE:


sage: f1 = lambda x: gv_bound_asymp(x,2)
sage: P1 = plot(f1,0,1/2,linestyle=":")
sage: f2 = lambda x: plotkin_bound_asymp(x,2)
sage: P2 = plot(f2,0,1/2,linestyle="--")
sage: f3 = lambda x: elias_bound_asymp(x,2)
sage: P3 = plot(f3,0,1/2,rgbcolor=(1,0,0))
sage: f4 = lambda x: singleton_bound_asymp(x,2)
sage: P4 = plot(f4,0,1/2,linestyle="-.")
sage: f5 = lambda x: mrrw1_bound_asymp(x,2)
sage: P5 = plot(f5,0,1/2,linestyle="steps")
sage: f6 = lambda x: hamming_bound_asymp(x,2)
sage: P6 = plot(f6,0,1/2,rgbcolor=(0,1,0))
sage: show(P1+P2+P3+P4+P5+P6)


If you are interested in computing what c might be in the above statement B(c,p), the following SAGE code may help.
First, the Python function below will return a random list of elements without repetition, if desired) from an object.

def random_elements(F,n,allow_repetition=True):
    """
    F is a SAGE object for which random_element is a method 
    and having >n elements. For example, a finite field. Here 
    n>1 is an integer. The output is to have no repetitions 
    if allow_repetion=False.
    EXAMPLES:
        sage: p = nth_prime(100)
        sage: K = GF(p)
        sage: random_elements(K,2,allow_repetition=False)
        [336, 186]
    The command random_elements(K,p+1,allow_repetition=False)
    triggers a ValueError.
    """
    if (n>F.order()-2 and allow_repetition==False):
        raise ValueError("Parameters are not well-choosen.")
    L = []
    while len(L)<n:
        c = F.random_element()
        if allow_repetition==False:
            if not(c in L):
                L = L+[c]
        else:
            L = L+[c]
    return L

Next, the SAGE code computes |X_S(GF(p))|/p for a particular choice of the parameters. This gives a lower bound for c.


sage: p = nth_prime(100)
sage: K = GF(p)
sage: R. = PolynomialRing(K, "x")
sage: S = random_elements(K,ZZ((p+1)/2),allow_repetition=False)
sage: fS = prod([x - a for a in S])
sage: XS = HyperellipticCurve(fS)
sage: RR(len(XS.points())/p)
0.975970425138632

Based on experiments with this code, it appears that for “random” chices of S, and p “large”, the quotient |X_S(GF(p))|/p is very close to 1.

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