Barker Code Calculator

Barker-13: + + + + + − − + + − + − +. Slide the code past itself and add the products of the chips that overlap. At shift 0 all 13 products are +1, so C(0) = 13. At shift 1 the 12 overlapping products add to 0, at shift 2 the 11 products add to 1, and the pattern carries on: C(1) to C(12) = 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1. The peak sidelobe level is 1, so the peak-to-sidelobe ratio is 20 × log10(13 ÷ 1) = 22.28 dB. The six sidelobes of 1 add to 6, so the merit factor is 13² ÷ (2 × 6) = 169 ÷ 12 = 14.08, and every |C(k)| ≤ 1 makes it a Barker sequence.

Common mistake 1: correlating the 0/1 bits directly. Written as bits, Barker-13 is 1111100110101. Multiplying the 0s and 1s as they are only counts overlapping 1s: C(0) = 9 and C(1) = C(2) = 5, which looks like a 5.11 dB code (20 × log10(9 ÷ 5)). Map the bits to chips first (1 → +1, 0 → −1; the opposite mapping gives the same result) and the sidelobes drop back to 0 and 1.

Common mistake 2: using the periodic instead of the aperiodic correlation. The 15-chip m-sequence 000100110101111 has a periodic autocorrelation of −1 at every nonzero shift, so if the code were repeated back to back it would look better than a Barker code. A radar pulse or a single sync word is sent once, and the matched filter then sees the aperiodic correlation: C(1) to C(14) = 0, 1, 2, 1, 0, 1, −2, 1, −2, −1, −2, −3, −2, −1. Its peak sidelobe is 3 at shift 12, giving 20 × log10(15 ÷ 3) = 13.98 dB and a merit factor of 225 ÷ 70 = 3.21. It is not a Barker sequence.

R. H. Barker described the sequences in 1953, in “Group synchronizing of binary digital systems” (in Communication Theory, edited by W. Jackson). He wanted a synchronising pattern whose correlator output was much larger when it lined up than at any partial overlap, gave examples of lengths 3, 7 and 11, and suspected that longer ones did not exist. Later engineers relaxed his condition to |C(k)| ≤ 1, which is the definition used today, and took the codes into pulse compression and radar, where a long coded pulse carries more energy while the matched filter still resolves a single chip.

Only seven lengths have Barker sequences: 2, 3, 4, 5, 7, 11 and 13. Lengths 2 and 4 each have two codes that are not reversed or negated copies of each other (+ + and + −; + + + − and + + − +), which is why the list above offers both. For odd lengths, R. Turyn and J. Storer proved in 1961 that none is longer than 13 (On binary sequences, Proc. American Mathematical Society 12, 394–399); K.-U. Schmidt and J. Willms published a shorter proof in 2016. For even lengths, Turyn and Storer pointed out that a Barker sequence would also have every periodic sidelobe equal to zero, and in 1965 Turyn showed that such a length must be 4u² for an odd u that is not a prime power, ruling out every even length from 14 to 12,099.

The even case appears to have been closed in 2026. A periodic sequence with all sidelobes zero is the first row of a circulant Hadamard matrix, and an AI-generated OpenAI preprint (The circulant Hadamard conjecture, 23 September 2026) claims that these exist only in orders 1 and 4. Its machine-checked Lean proof, published with Lean Comparator checks, covers that classification and the even Barker lengths, 2 and 4; the odd lengths rest on the 1961 theorem, which the formal statement does not include. The result has not yet been peer reviewed. See what “Lean-checked” means.

Where a Barker code is too short, engineers use codes with the lowest peak sidelobe found by exhaustive search. A. N. Leukhin and E. N. Potekhin list them up to length 68: a peak sidelobe of 2 is possible at every non-Barker length up to 21 and again at 25 and 28, and 3 is the best possible from 29 to 48.

Sources: J. Jedwab, “What can be used instead of a Barker sequence?” (Contemporary Mathematics 461, 2008); K.-U. Schmidt and J. Willms, “Barker sequences of odd length” (Designs, Codes and Cryptography 80, 2016); OpenAI, Lean scope note for result 179; A. N. Leukhin and E. N. Potekhin, “Binary sequences with minimum peak sidelobe level up to length 68”; IEEE Std 802.11, 1999 Edition, subclause 15.4.6.3.

This calculator computes the full aperiodic autocorrelation of any binary code up to 4,096 chips and reports its peak sidelobe level, peak-to-sidelobe ratio, integrated sidelobe level, merit factor and whether it is a Barker sequence. It reads + and − signs, ±1 values or 0/1 bits, and can add the periodic autocorrelation for a code that repeats. The plot shows the chips and the matched-filter output with the ±1 Barker band marked, and the zoom brings small sidelobes into view next to a tall peak.

The exhaustive search checks every sequence of each length from 2 to 28 chips, using branch and bound so whole families of sequences are ruled out at once, and reports the lowest peak sidelobe possible, how many sequences reach it, and how many are Barker sequences. It runs in a background thread, so the page stays responsive and you can stop it at any time. Everything runs in your browser; nothing you enter is sent to a server.

It is for radar, sonar and radio engineers choosing a pulse-compression or spreading code, for anyone checking a sync word, and for students who want to see for themselves that no Barker code exists at lengths such as 6, 8, 9, 10 and 12.

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