—   length ($n\geq2$: the normalization $a_0=a_1=1$ needs two terms)

—   normalized Barker sequence (a word in + and - of length $n$, where + means $1$ and - means $-1$, normalized by $a_0=a_1=1$)

If $B_a(x)=\sum_{j=0}^{n-1}a_jx^j$, then $B_a(x)x^{n-1}B_a(1/x)=\sum_{k=-(n-1)}^{n-1}c_{|k|}x^{n-1+k}$, where $c_k=\sum_{j=0}^{n-1-k}a_ja_{j+k}$ and $c_0=n$.

The aperiodic merit factor of the sequence is $F=n^2/(2\sum_{k=1}^{n-1}c_k^2)$. For the length-$13$ sequence, $F=169/12$.

Negating every term and replacing $a_j$ by $(-1)^ja_j$ preserve the magnitudes of the aperiodic autocorrelations, so a nontrivial Barker sequence can be represented with $a_0=a_1=1$. This table stores the normalized sign words listed by Borwein and Mossinghoff [1].

Coefficients run low degree first: the sign in position $j$ of the sequence is the coefficient of $x^j$. With this indexing, $c_0=n$.

Apart from the trivial sequence of length $1$, Barker sequences are known only for lengths $2,3,4,5,7,11,13$, and it is conjectured that no longer Barker sequence exists [2]. The nonexistence is proved for odd lengths greater than $13$, while the existence of an even Barker sequence longer than $4$ is open.

Sage

import numberdb.sage as numberdb
from sage.rings.integer_ring import ZZ
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing

R = PolynomialRing(ZZ, "x")
x = R.gen()

def barker_polynomial(signs):
    return sum((ZZ(1) if sign == "+" else ZZ(-1)) * x**j
               for j, sign in enumerate(signs))

barker_polynomial("+++++--++-+-+")

Peter Borwein and Michael J. Mossinghoff, Barker sequences and flat polynomials, in Number Theory and Polynomials, London Mathematical Society Lecture Note Series 352, Cambridge University Press, 2008, 71-88. (doi) (MR)

J. Willms, A note on Barker sequences of even length, 2021. (arXiv) (doi)

MathWorld: Barker Code

Shapiro polynomials $P_n$ —   use the Rudin-Shapiro sign rule for a family of binary sequences with small aperiodic autocorrelations

Complementary Shapiro polynomials $Q_n$ —   form the second polynomial sequence in the Rudin-Shapiro pair, another family of binary sequences with small aperiodic autocorrelations

Merit factors of the Legendre sequences —   stores exact aperiodic merit factors for another binary-sequence family, rather than the sign polynomials themselves

Fekete polynomials $f_p$ —   use Legendre-symbol coefficients, with zero constant term, rather than Barker sign words normalized by their first two signs

Entries are of type: integral polynomial

How well the digits are known: exact

Table is complete: no (it holds every normalized Barker sequence presently known, namely the eight sequences listed by [1], whose lengths are $2,3,4,4,5,7,11,13$)

How they were obtained:

The values are exact polynomials over $\mathbb Z$.

more

The generator builds each polynomial from the sign word listed by [1] and checks the Barker autocorrelation condition, the autocorrelation polynomial identity in (1), the aperiodic merit factor in (2), and the source sign words against a brute-force enumeration of all normalized sign words through length $13$.