MacWilliams, F. J.; Sloane, N. J. A. The theory of error-correcting codes. Parts I, II. (English) Zbl 0369.94008 North-Holland Mathematical Library. Vol. 16. Amsterdam-New York-Oxford: North-Holland Publishing Company. Part I: xv, 369 p. $ 24.50; Dfl. 60.00. Part II: ix, 391 p. $ 32.75; Dfl. 80.00. Set of two Vols. $ 50.95; Dfl. 125.00 (1977). Mit diesem zweibändigen Werk haben die Autoren eine umfassende und gründliche Darstellung wesentlicher Ergebnisse und Forschungsrichtungen im Bereich der Codierungstheorie vorgelegt. Schon aus der Übersicht über die Kapitelüberschriften kann man einen Eindruck von der Inhaltsfülle gewinnen: 1. lineare Codes, 2. nichtlineare Codes, Hadamardmatrizen, Blockpläne, Golay-Code, 3. Einführung von BCH-Codes, 4. endliche Körper, 5. duale Codes, Gewichtsverteilungen, 6. Blockpläne, perfekte Codes, 7. zyklische Codes, 8. zyklische Codes, Idempotente, Mattson-Solomon-Polynome, 9. BCH-Codes, 10. Reed-Solomon-Codes, Justesen-Codes, 11. Codes mit maximaler Minimaldistanz, 12. Alternanten-Codes, Goppa-Codes, Srivastava-Codes und verwandte, 13.–15. Reed-Muller-Codes, Kerdock-Codes, Preparata-Codes, 16. Codes aus quadratischen Resten, 17. Qualitätsschranken, 18. Operationen auf Codes, 19. selbstduale Codes, Invariantentheorie, 20. Golay-Codes, 21. Assoziationsschemata; dazu in Anhängen: Tabelle der besten bekannten Codes, einiges über endliche Geometrien, eine umfangreiche Bibliographie mit 1478 Titeln.Schon diese Liste läßt erkennen, daß dieses Werk die bisherigen Bücher über Codierungstheorie hervorragend ergänzt. Beim Lesen kann man dann feststellen, daß selbst in “klassischen” Bereichen (z.B. BCH-Codes, zyklische Codes) vieles Neue und viel Bekanntes in neuer Form geboten wird. Etliches erscheint hier allerdings auch erstmals in dieser Ausführlichkeit in diesem Buch. Hervorzuheben sind: Die Ergebnisse über Gewichtsverteilungen (großenteils auf die Autoren selbst zurückgehend); der Zusammenhang zwischen kombinatorischen und geometrischen Strukturen wie z.B. Blockplänen einerseits und Codes andererseits; die Zusammenfassung von z.B. BCH-, Goppa- und Srivastava-Codes zu den Alternantencodes; die Qualitätsschranken aus Überlegungen der linearen Programmierung; detaillierte Beschreibung der Golay-Codes; Anwendung der Assoziationsschemata im Bereich der Codierungstheorie (auf Delsarte zurückgehend). Dies sind nur einige der eindrucksvollsten Punkte. We schon gesagt, das Neue geht bis in die Details der Darstellung.Das Werk ist sicher keine Lektüre für Anfänger, aber dem in diesem Bereich Forschenden oder Lehrenden bietet es viel. Es ist auf seine immerhin etwa 750 Seiten durchgängig nie trocken geschrieben. Der benötigte Hintergrund aus anderen mathematischen Gebieten ist beträchtlich und vielfältig, wird aber durch klug gesetzte Hinweise häufig schnell erläutert.Der Referent hat die Gelegenheit genutzt, das Werk bei der Vorbereitung der Lehrveranstaltung zu Rate zu ziehen: Es eignet sich sehr gut als Vorlesungsgrundlage.Den Autoren ist ein hervorragend brauchbares Werk gelungen. Reviewer: H. Jürgensen Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 30 ReviewsCited in 2601 Documents MSC: 94Bxx Theory of error-correcting codes and error-detecting codes 94-02 Research exposition (monographs, survey articles) pertaining to information and communication theory 05Bxx Designs and configurations 11T71 Algebraic coding theory; cryptography (number-theoretic aspects) Keywords:linear codes; nonlinear codes; Hadamard matrices; block diagrams; Golay code; BCH codes; finite fields; dual codes; weight distributions; perfect codes; cyclic codes; idempotents; Mattson-Solomon polynomials; Reed-Solomon codes; Justesen codes; codes with maximum minimal distance; alternant codes; Goppa codes; Srivastava codes; Reed-Muller codes; Kerdock codes; Preparata codes; quadratic residue codes; quality bounds; operations on codes; self-dual codes; invariant theory; Golay codes; association schemes; table of best codes; finite geometries; Bibliography × Cite Format Result Cite Review PDF Online Encyclopedia of Integer Sequences: Numbers k == 2 (mod 4) that are the orders of conference matrices. Number of equivalence classes of Boolean functions modulo linear functions. Leech triangle: k-th number (0 <= k <= n) in n-th row (0 <= n) is number of octads in S(5,8,24) containing k given points and missing n-k given points. Triangle in which k-th number (0<=k<=n) in n-th row (0<=n) is number of dodecads in Golay code G_24 containing k given points and missing n-k given points. Triangle in which k-th number (0<=k<=n) in n-th row (0<=n) is number of hexads in S(5,6,12) containing k given points and missing n-k given points. Weight distribution of binary Golay code of length 24. Weight distribution of binary [ 48,24,12 ] quadratic residue code. Weight distribution of ternary [ 24,12,9 ] quadratic residue code (also of Pless symmetry code). Weight distribution of [ 64,22,16 ] 2nd-order Reed-Muller code of length 64. Weight distribution of [ 23,12,7 ] binary perfect Golay code. Weight distribution of [8,4,4] Hamming code. Weight distribution of [8,4,4] Hamming code. Weight distribution of [ 7,4,3 ] Hamming code. Weight distribution of Karlin’s [28,14,8] double circulant code. Hadamard maximal determinant problem: largest determinant of a (real) {0,1}-matrix of order n. Number of bent functions of 2n variables. The coding-theoretic function A(n,4). The coding-theoretic function A(n,6). The coding-theoretic function A(n,8). Weight distribution of [ 128,29,32 ] 2nd-order Reed-Muller code. Sum of Gaussian binomial coefficients [n,k] for q=2 and k=0..n. Number of cyclotomic cosets of 2 mod 2n+1. Triangle read by rows of partial sums of binomial coefficients: T(n,k) = Sum_{i=0..k} binomial(n,i) (0 <= k <= n); also dimensions of Reed-Muller codes. Weight distribution of any one of the five doubly-even binary [32,16,8] codes (quadratic residue, Reed-Muller, etc.). Weight distribution of binary (16,256,6) nonlinear Nordstrom-Robinson code. Weight distribution of [16,11,4] extended Hamming code of length 16. Weight distribution of extended Hamming code of length 32 (or 3rd-order Reed-Muller code). Weight distribution of extended Hamming code of length 64. Weight distribution of [128,120,4] extended Hamming code of length 128. This is also the Reed-Muller code RM(5,7). Weight distribution of extended Hamming code of length 256. Weight distribution of [15,11,3] Hamming code of length 15 and minimal distance 3. Weight distribution of d=3 Hamming code of length 31. Weight distribution of d=3 Hamming code of length 63. Weight distribution of d=3 Hamming code of length 127. Weight distribution of [255,247,3] Hamming code of length 255. Weight distribution of d=4 Hamming code of length 15. Weight distribution of d=4 Hamming code of length 31. Weight distribution of d=4 Hamming code of length 63. Weight distribution of d=4 Hamming code of length 127. Weight distribution of d=4 Hamming code of length 255. Period 3: repeat [0, 1, 1]. A binary m-sequence: expansion of reciprocal of x^3 + x^2 + 1 (mod 2), shifted by 2 initial 0’s. A binary m-sequence: expansion of reciprocal of x^3 + x + 1 (mod 2, shifted by 2 initial 0’s). A binary m-sequence: expansion of reciprocal of x^4+x+1. A binary m-sequence: expansion of reciprocal of x^5+x^4+x^2+x+1. A binary m-sequence: expansion of reciprocal of x^5+x^3+x^2+x+1. A binary m-sequence: expansion of reciprocal of x^5 + x^2 + 1. A binary m-sequence: expansion of reciprocal of x^5+x^4+x^3+x+1. A binary m-sequence: expansion of reciprocal of x^5+x^3+1. A binary m-sequence: expansion of the reciprocal of x^5+x^4+x^3+x^2+1. A binary m-sequence: expansion of reciprocal of x^6+x^5+x^4+x+1. A binary m-sequence: expansion of reciprocal of x^6+x^5+x^3+x^2+1. A binary m-sequence: expansion of reciprocal of x^6+x^5+x^2+x+1. A binary m-sequence: expansion of reciprocal of x^6+x+1. A binary m-sequence: expansion of reciprocal of x^6+x^4+x^3+x+1. A binary m-sequence: expansion of reciprocal of x^6+x^5+x^4+x^2+1. A binary m-sequence: expansion of reciprocal of x^6+x^5+1. A binary m-sequence: expansion of reciprocal of x^7+x^6+x^5+x^4+x^3+x^2+1. A binary m-sequence: expansion of reciprocal of x^7+x^4+1. A binary m-sequence: expansion of reciprocal of x^7+x^6+x^4+x^2+1. A binary m-sequence: expansion of reciprocal of x^7+x^5+x^2+x+1. A binary m-sequence: expansion of reciprocal of x^7+x^5+x^3+x+1. A binary m-sequence: expansion of reciprocal of x^7+x^6+x^4+x+1. A binary m-sequence: expansion of reciprocal of x^7+x^6+x^5+x^4+x^2+x+1. A binary m-sequence: expansion of reciprocal of x^7+x^6+x^5+x^3+x^2+x+1. A binary m-sequence: expansion of reciprocal of x^7+x+1. A binary m-sequence: expansion of reciprocal of x^7+x^5+x^4+x^3+x^2+x+1. A binary m-sequence: expansion of reciprocal of x^7+x^4+x^3+x^2+1. A binary m-sequence: expansion of reciprocal of x^7+x^6+x^3+x+1. A binary m-sequence: expansion of reciprocal of x^7 + x^6 + 1. A binary m-sequence: expansion of reciprocal of x^7 + x^6 + x^5 + x^4 + 1. A binary m-sequence: expansion of reciprocal of x^7+x^5+x^4+x^3+1. A binary m-sequence: expansion of reciprocal of x^7+x^3+x^2+x+1. A binary m-sequence: expansion of reciprocal of x^7+x^3+1. A binary m-sequence: expansion of reciprocal of x^7+x^6+x^5+x^2+1. A binary m-sequence: expansion of reciprocal of x^8+x^6+x^4+x^3+x^2+x+1. A binary m-sequence: expansion of reciprocal of x^8+x^5+x^4+x^3+1. A binary m-sequence: expansion of reciprocal of x^8+x^7+x^5+x^3+1. A binary m-sequence: expansion of reciprocal of x^8+x^7+x^6+x^5+x^4+x^2+1. A binary m-sequence: expansion of reciprocal of x^8+x^7+x^6+x^5+x^4+x^3+1. A binary m-sequence: expansion of reciprocal of x^8+x^4+x^3+x^2+1. A binary m-sequence: expansion of reciprocal of x^8+x^6+x^5+x^4+x^2+x+1. A binary m-sequence: expansion of reciprocal of x^8+x^7+x^5+x+1. A binary m-sequence: expansion of reciprocal of x^8+x^7+x^3+x+1. A binary m-sequence: expansion of reciprocal of x^8+x^5+x^4+x^3+x^2+x+1. A binary m-sequence: expansion of reciprocal of x^8+x^7+x^5+x^4+x^3+x^2+1. A binary m-sequence: expansion of reciprocal of x^8+x^7+x^6+x^4+x^3+x^2+1. A binary m-sequence: expansion of reciprocal of x^8+x^6+x^3+x^2+1. A binary m-sequence: expansion of reciprocal of x^8+x^7+x^3+x^2+1. A binary m-sequence: expansion of reciprocal of x^8+x^6+x^5+x^2+1. A binary m-sequence: expansion of reciprocal of x^8+x^7+x^6+x^4+x^2+x+1. A binary m-sequence: expansion of reciprocal of x^8+x^7+x^6+x^3+x^2+x+1. A binary m-sequence: expansion of reciprocal of x^8+x^7+x^2+x+1. A binary m-sequence: expansion of reciprocal of x^8+x^7+x^6+x+1. A binary m-sequence: expansion of reciprocal of x^8+x^7+x^6+x^5+x^2+x+1. A binary m-sequence: expansion of reciprocal of x^8+x^7+x^5+x^4+1. A binary m-sequence: expansion of reciprocal of x^8+x^6+x^5+x+1. A binary m-sequence: expansion of reciprocal of x^8+x^4+x^3+x+1. A binary m-sequence: expansion of reciprocal of x^8+x^6+x^5+x^4+1. A binary m-sequence: expansion of reciprocal of x^8+x^7+x^6+x^5+x^4+x+1. A binary m-sequence: expansion of reciprocal of x^8+x^5+x^3+x^2+1. A binary m-sequence: expansion of reciprocal of x^8+x^6+x^5+x^4+x^3+x+1. A binary m-sequence: expansion of reciprocal of x^8+x^5+x^3+x+1. A binary m-sequence: expansion of reciprocal of x^8+x^7+x^4+x^3+x^2+x+1. A binary m-sequence: expansion of reciprocal of x^8+x^6+x^5+x^3+1. A binary m-sequence: expansion of reciprocal of x^9+x^4+1. A binary m-sequence: expansion of reciprocal of x^10+x^3+1. A binary m-sequence: expansion of reciprocal of x^11 + x^2 + 1 (mod 2, shifted by 10 initial 0’s). A binary m-sequence: expansion of reciprocal of x^12+x^7+x^4+x^3+1. A binary m-sequence: expansion of reciprocal of x^13+x^4+x^3+x+1. A binary m-sequence: expansion of reciprocal of x^14 + x^12 + x^11 + x + 1 (mod 2, shifted by 13 initial 0’s). A binary m-sequence: expansion of reciprocal of x^15+x+1. A binary m-sequence: expansion of reciprocal of x^16+x^5+x^3+x^2+1. A binary m-sequence: expansion of reciprocal of x^17+x^3+1. A binary m-sequence: expansion of reciprocal of x^18 + x^7 + 1 (mod 2, shifted by 17 initial 0’s). A binary m-sequence: expansion of reciprocal of x^19 + x^6 + x^5 + x + 1 (mod 2, shifted by 18 initial 0’s). A binary m-sequence: expansion of reciprocal of x^20 + x^3 + 1 (mod 2, shifted by 19 initial 0’s). A binary m-sequence: expansion of reciprocal of x^21 + x^2 + 1 (mod 2, shifted by 20 initial 0’s). A binary m-sequence: expansion of reciprocal of x^22 + x + 1 (mod 2, shifted by 21 initial 0’s). A binary m-sequence: expansion of reciprocal of x^23 + x^5 + 1 (mod 2, shifted by 22 initial 0’s). A binary m-sequence: expansion of reciprocal of x^24 + x^4 + x^3 + x + 1 (mod 2, shifted by 23 initial 0’s). A binary m-sequence: expansion of reciprocal of x^25 + x^3 + 1 (mod 2, shifted by 24 initial 0’s). A binary m-sequence: expansion of reciprocal of x^26 + x^8 + x^7 + x + 1 (mod 2, shifted by 25 initial 0’s). A binary m-sequence: expansion of reciprocal of x^27 + x^8 + x^7 + x + 1 (mod 2, shifted by 26 initial 0’s). A binary m-sequence: expansion of reciprocal of x^28 + x^3 + 1 (mod 2, shifted by 27 initial 0’s). A binary m-sequence: expansion of reciprocal of x^29 + x^2 + 1 (mod 2, shifted by 28 initial 0’s). A binary m-sequence: expansion of reciprocal of x^30 + x^16 + x^15 + x + 1 (mod 2, shifted by 29 initial 0’s). A binary m-sequence: expansion of reciprocal of x^31 + x^3 + 1 (mod 2, shifted by 30 initial 0’s). A binary m-sequence: expansion of reciprocal of x^32 + x^28 + x^27 + x + 1 (mod 2, shifted by 31 initial 0’s). Triangle of Gaussian binomial coefficients (or q-binomial coefficients) [n,k] for q = 2. Triangle of Gaussian binomial coefficients [ n,k ] for q = 3. Triangle of Gaussian binomial coefficients [ n,k ] for q = 4. Triangle of Gaussian binomial coefficients [ n,k ] for q = 5. Triangle of Gaussian binomial coefficients [ n,k ] for q = 6. Triangle of Gaussian binomial coefficients [ n,k ] for q = 7. Triangle of Gaussian binomial coefficients [ n,k ] for q = 8. Triangle of Gaussian binomial coefficients [ n,k ] for q = 9. Triangle of Gaussian binomial coefficients [ n,k ] for q = 10. Triangle of Gaussian binomial coefficients [ n,k ] for q = 11. Triangle of Gaussian binomial coefficients [ n,k ] for q = 12. Triangle of Gaussian binomial coefficients [ n,k ] for q = 13. Triangle of Gaussian binomial coefficients [ n,k ] for q = 14. Triangle of Gaussian binomial coefficients [ n,k ] for q = 15. Triangle of Gaussian binomial coefficients [ n,k ] for q = 16. Triangle of Gaussian binomial coefficients [ n,k ] for q = 17. Triangle of Gaussian binomial coefficients [ n,k ] for q = 18. Triangle of Gaussian binomial coefficients [ n,k ] for q = 19. Triangle of Gaussian binomial coefficients [ n,k ] for q = 20. Triangle of Gaussian binomial coefficients [ n,k ] for q = 21. Triangle of Gaussian binomial coefficients [ n,k ] for q = 22. Triangle of Gaussian binomial coefficients [ n,k ] for q = 23. Triangle of Gaussian binomial coefficients [ n,k ] for q = 24. Gaussian binomial coefficients [ n,10 ] for q = 2. Gaussian binomial coefficients [ n,11 ] for q = 2. Gaussian binomial coefficients [ n,12 ] for q = 2. Gaussian binomial coefficients [ n,7 ] for q = 3. Gaussian binomial coefficients [ n,8 ] for q = 3. Gaussian binomial coefficients [ n,9 ] for q = 3. Gaussian binomial coefficients [ n,10 ] for q = 3. Gaussian binomial coefficients [ n,11 ] for q = 3. Gaussian binomial coefficients [ n,12 ] for q = 3. Gaussian binomial coefficients [ n,6 ] for q = 4. Gaussian binomial coefficients [ n,7 ] for q = 4. Gaussian binomial coefficients [ n,8 ] for q = 4. Gaussian binomial coefficients [ n,9 ] for q = 4. Gaussian binomial coefficients [ n,10 ] for q = 4. Gaussian binomial coefficients [ n,11 ] for q = 4. Gaussian binomial coefficients [ n,12 ] for q = 4. Gaussian binomial coefficients [ n,5 ] for q = 5. Gaussian binomial coefficients [ n,6 ] for q = 5. Gaussian binomial coefficients [ n,7 ] for q = 5. Gaussian binomial coefficients [ n,8 ] for q = 5. Gaussian binomial coefficients [ n,9 ] for q = 5. Gaussian binomial coefficients [ n,10 ] for q = 5. Gaussian binomial coefficients [ n,11 ] for q = 5. Gaussian binomial coefficients [ n,12 ] for q = 5. Gaussian binomial coefficients [ n,5 ] for q = 6. Gaussian binomial coefficients [ n,6 ] for q = 6. Gaussian binomial coefficients [ n,7 ] for q = 6. Gaussian binomial coefficients [ n,8 ] for q = 6. Gaussian binomial coefficients [ n,9 ] for q = 6. Gaussian binomial coefficients [ n,10 ] for q = 6. Gaussian binomial coefficients [ n,11 ] for q = 6. Gaussian binomial coefficients [ n,12 ] for q = 6. Gaussian binomial coefficients [ n,4 ] for q = 7. Gaussian binomial coefficients [ n,5 ] for q = 7. Gaussian binomial coefficients [ n,6 ] for q = 7. Gaussian binomial coefficients [ n,7 ] for q = 7. Gaussian binomial coefficients [ n,8 ] for q = 7. Gaussian binomial coefficients [ n,9 ] for q = 7. Gaussian binomial coefficients [ n,10 ] for q = 7. Gaussian binomial coefficients [ n,11 ] for q = 7. Gaussian binomial coefficients [ n,12 ] for q = 7. Gaussian binomial coefficients [ n,4 ] for q = 8. Gaussian binomial coefficients [ n,5 ] for q = 8. Gaussian binomial coefficients [ n,6 ] for q = 8. Gaussian binomial coefficients [ n,7 ] for q = 8. Gaussian binomial coefficients [ n,8 ] for q = 8. Gaussian binomial coefficients [ n,9 ] for q = 8. Gaussian binomial coefficients [ n,10 ] for q = 8. Gaussian binomial coefficients [ n,11 ] for q = 8. Gaussian binomial coefficients [ n,12 ] for q = 8. Gaussian binomial coefficients [ n,4 ] for q = 9. Gaussian binomial coefficients [ n,5 ] for q = 9. Gaussian binomial coefficients [ n,6 ] for q = 9. Gaussian binomial coefficients [ n,7 ] for q = 9. Gaussian binomial coefficients [ n,8 ] for q = 9. Gaussian binomial coefficients [ n,9 ] for q = 9. Gaussian binomial coefficients [ n,10 ] for q = 9. Gaussian binomial coefficients [ n,11 ] for q = 9. Gaussian binomial coefficients [ n,12 ] for q = 9. Weight distribution of [ 17,8,6 ] binary quadratic-residue code. Weight distribution of (64,4096,28) Kerdock code. Weight distribution of (64, 2^52, 6) Preparata code. Weight distribution of (256,2^16,120) Kerdock code. Weight distribution of (256,2^240,6) Preparata code. Total number of self-dual binary codes of length 2n. Totally isotropic spaces of index n in symplectic geometry of dimension 2n. Total number of doubly-even self-dual binary codes of length 8n. Weight distribution of [ 17,9,5 ] binary quadratic-residue code. Weight distribution of [ 31,16,7 ] binary BCH and quadratic-residue codes. Weight distribution of [ 31,15,8 ] binary quadratic-residue code. Weight distribution of [ 47,24,11 ] binary quadratic-residue code. Weight distribution of [ 47,23,12 ] binary quadratic-residue code. Leading term in extremal weight enumerator of doubly-even binary self-dual code of length 24n. Second term in extremal weight enumerator of doubly-even binary self-dual code of length 24n. Number of cyclotomic cosets C of 2 mod 2n+1 such that -C = C. Number of cyclotomic cosets C of 2 mod 2n+1 such that -C is not equal to C, divided by 2. Number of prime powers <= n. Table in which n-th row gives exponents (in decreasing order) of lexicographically earliest primitive irreducible polynomial of degree n over GF(2). n is congruent to 1 (mod 4) and is not the sum of two squares. Weight enumerator of [16,7,6] extended BCH code. Johnson bound J(n,4,2). Hamming distance between n and A102370(n) (in binary). Highest minimal distance of any Type I (strictly) singly-even binary self-dual code of length 2n. Highest minimal distance of any Type II doubly-even binary self-dual code of length 8n. Highest minimal Hamming distance of any Type 3 ternary self-dual code of length 4n. Weight distribution of [12,6,6]_3 ternary extended Golay code. Weight distribution of [11,6,5]_3 ternary Golay perfect code. Number of inequivalent codes attaining highest minimal distance of any Type I (strictly) singly-even binary self-dual code of length 2n. Weight distribution of [13,6,6]_3 ternary code p_{13}. Weight distribution of quadratic residue code of length 23 over GF(3). Weight distribution of [72,36,16] doubly-even binary self-dual extended quadratic-residue (or QR) code. G.f. 1/( (1 + x)^7*(1 -7*x +28*x^2 -84*x^3 +210*x^4 -462*x^5 +924*x^6 -1463*x^7 +1738*x^8 -1463*x^9 +924*x^10 -462*x^11 +210*x^12 -84*x^13 +28*x^14 -7*x^15 +x^16) ) Characteristic polynomials of a binomial modulo two Hadamard transpose general matrix: t(n,m,d) = If[ m <= n, binomial(n, m) mod 2], 0]; M(d)=t(n,m,d).Transpose[t(n,m,d)]. A triangle of coefficients Pseudo-Hadamard matrices as integer characteristic polynomials (the code and initial values are very long, but the basic recurrence is the Hadamard matrix self-similarity). Consider binary linear [N,K,D] codes with D=6 and redundancy R = N-K = n; a(n) = maximal value of N. Numbers n such that A008949(n) is a power of 2. Smallest integer t such that 1 + binomial(n,1) + binomial(n,2) + ... + binomial(n,t) is a power of 2. The Griesmer lower bound q_4(5,n) on the length of a linear code over GF(4) of dimension 5 and minimal distance n. Maximal gap between quadratic residues mod n. Maximal gap between quadratic residues mod n; here quadratic residues must be coprime to n. Number of Type I (singly-even) self-dual binary codes of length 2n.