Hamming Codes with error detection and Correction Part-1

Duration: 12 min

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Module outline

  1. About the Course: About the Course
  2. Section 1 | Language (Hindi): भाषा, लिपि और वर्ण, वर्तनी, वर्ण-विन्यास, अनुस्वार, अनुनासिक, चंद्रबिंदु और ‘र’ के रूप, शब्द, पद और शब्द-भेद, संज्ञा और सर्वनाम, पद-भेद और रूप-विकार, लिंग, वचन, पुरुष और कारक, काल, वाच्य, वाक्य और वाक्य-भेद, संधि, समास, विलोम शब्द, पर्यायवाची शब्द, सामान्य अशुद्धियाँ, अनेक शब्दों के लिए एक शब्द, शब्द-भंडार, अपठित बोध, Multi-Topic Content (Cross-Concept Material), वर्णमाला
  3. Section 1 | Language (उर्दू भाषा — 22 प्रश्न (निश्चित)): رسم الخط، حروفِ تہجی اور املا / Script, Alphabet and Spelling, اسم، ضمیر اور صفت / Nouns, Pronouns and Adjectives, الفاظ، محاورے اور ضرب الامثال / Vocabulary, Idioms and Proverbs, تفہیمِ عبارت، انشا اور ترجمہ / Comprehension, Writing and Translation, Comprehension
  4. Section 1 | Language (বাংলা / बांग्ला भाषा — 22 प्रश्न (निश्चित)): শব্দার্থ, বাগধারা ও ভাষাশুদ্ধি / Meaning, Idioms and Correct Usage, বাক্য ও বাচ্য / Sentences and Voice, শব্দগঠন ও শব্দভান্ডার / Word Formation and Vocabulary, অনুবাদ, বোধপরীক্ষণ ও রচনা / Translation, Comprehension and Composition
  5. Section 1 | Language (English): Vocabulary, Noun, Subject Verb Agreement (Verb Noun Agreement), Adjectives, Verbs and Tenses, Pronoun, Verb, Adverb, Prepositions and Conjunctions, Parts of Speech, Determiner, Conjunction, Modals, Transformation of Sentences, Direct and Indirect Speech, Sentence Correction (Error Correction), Sentence Completion (Fill in the blanks), Sentence Construction, Sentence Structure and Common Errors, Reading Comprehension, Cloze Test (Cloze Passage), Verbal Analogy (Word Based Analogy), Facts, Inference and Judgements
  6. Section 2 | Primary Maths & Reasoning: अनुपात और समानुपात / Ratio and Proportion, Divisibility Rules, मिश्रण और एलिगेशन / Mixture and Alligation, Permutation and Combination, संख्या पद्धति / Number System, महत्तम समापवर्तक और लघुत्तम समापवर्त्य / HCF and LCM, Sequence and Series (Series), समय, चाल और दूरी / Time, Speed and Distance, श्रेणी / Series, कूट लेखन एवं कूट वाचन / Coding-Decoding, अशाब्दिक तर्क / Non-Verbal Reasoning, प्रतिशत / Percentage, औसत / Average, लाभ, हानि और बट्टा / Profit, Loss and Discount, साधारण और चक्रवृद्धि ब्याज / Simple and Compound Interest, समय और कार्य / Time and Work, घन क्षेत्रमिति / Solid Mensuration, Age Problems, Arithmetic, सरलीकरण, घातांक और करणी / Simplification, Indices and Surds, Algebra, Probability, Trigonometry, Syllogisms, बैठक व्यवस्था एवं पहेलियाँ / Seating Arrangement and Puzzles, रक्त संबंध / Blood Relations, दिशा एवं दूरी / Direction and Distance, Analytical Reasoning (Counting Figures Reasoning), गणितीय संक्रियाएँ एवं लुप्त संख्या / Mathematical Operations and Missing Numbers, कैलेंडर एवं घड़ी / Calendar and Clock, Clock, घन, पासा एवं आकृतियों की गिनती / Cubes, Dice and Figure Counting, वेन आरेख, न्याय वाक्य एवं असमानता / Venn Diagrams, Syllogism and Inequalities, Data Arrangement, Inequality, सादृश्यता एवं वर्गीकरण / Analogy and Classification, क्रम एवं रैंकिंग / Order and Ranking, समुच्चय, संबंध, फलन और विविक्त गणित / Sets, Relations, Functions and Discrete Mathematics, संख्या पद्धति और संख्या सिद्धांत / Number Systems and Number Theory, निर्देशांक ज्यामिति, सदिश और त्रिविमीय ज्यामिति / Coordinate Geometry, Vectors and 3D Geometry, बहुपद, समीकरण और सम्मिश्र संख्याएँ / Polynomials, Equations and Complex Numbers
  7. Section 2 | General Awareness (GK + Current Affairs): Polity, वैज्ञानिक उपलब्धियाँ और अनुसंधान / Scientific Achievements and Research, Biology, राज्य विशेष सामान्य ज्ञान / State-Specific General Knowledge, Chemistry, History, Geography, Indian Economy, Environment, समसामयिकी / Current Affairs, Static General Knowledge
  8. Section 3 | Computer System and Org. (CSO): Basics of COA, Floating Point Rep, Cache Memory Organization, Instr Formats & Modes, Programming the Basic Computer, Input Output Organisation, Register Transfer and Microoperations, Control Unit Design
  9. Section 3 | Database Management System: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, SQL, File Organization & Indexing, Relational Algebra, Relational Calculus, Transaction Management, Concurrency Control
  10. Section 3 | Operating System: Introduction to OS, Process Management, CPU Scheduling, Threads & Process Creation, Memory Management, Virtual Memory, Disc Scheduling, File Management, Process Synchronization
  11. Section 3 | Computer Networks: Introduction to CN, Data Communication, DLL: Access Control, DLL: Flow Control, DLL: Error Control, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Transport Layer Services, TL: Congestion & UDP, Application Layer, Hardware basics
  12. Section 3 | Data Structures: Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing, Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Minimum Spanning Trees, Shortest Path Algos
  13. Section 3 | Theory Of Computation & Compiler Design: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines, Intro to Compilers, Lexical Analysis, Grammar & CFG, Syntax Analysis: Top-Down, Syntax Analysis: Bottom-Up, Semantic Analysis & SDT, Intermediate Code Gen, Code Optimization
  14. Section 3 | Emerging Trends (AI, Societal Ethics & MS Office): कंप्यूटर के मूल सिद्धांत / Computer Fundamentals, Emerging Tech Trends, Society, Law & Ethics, MS Office Applications, व्यापार, विपणन एवं उपभोक्ता संरक्षण / Trade, Marketing and Consumer Protection, साइबर सुरक्षा / Cyber Safety and Security, कंप्यूटर सॉफ़्टवेयर / Computer Software, ऑपरेटिंग सिस्टम / Operating System, कंप्यूटर नेटवर्क / Computer Networks, वर्ड प्रोसेसिंग (MS Word) / Word Processing (MS Word), Terms & Abbreviations, प्रेज़ेंटेशन (MS PowerPoint) / Presentation Software (MS PowerPoint)
  15. Section 3 | Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Maintenance and Configuration Management, Software Testing
  16. Section 3 | C Programming: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling
  17. Section 3 | OOPS, C++, JAVA, HTML: OOPS with C++, Java, HTML Basics, CSS
  18. Section 3 | Java: Java Fundamentals, Array, String, IO & References, OOPS with Java
  19. Section 3 | Python: Introduction to Python, Tokens, Variables, Data Types, I/O, Operators & Expressions, Control Flow & Looping, Collections & Strings, Functions & Modules, File Handling & Data Analysis
  20. Section 3 | Digital Electronics: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation
  21. Section 3 | Multimedia: Multimedia
  22. BPSC Previous Year Papers:
  23. BPSC Full Mock Tests:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This lecture introduces Hamming codes as single-bit error-correcting block codes with minimum distance dmin = 3, enabling detection of up to two errors or correction of one error. The instructor defines the relationship between total codeword length n, data bits k, and redundancy r using n = 2^r - 1 and k = n - r or k = 2^r - r - 1. A table maps bit positions 1 through 20 to parity bits at powers of two (p1, p2, p4, p8, p16) and data bits elsewhere, with X marks showing which positions each parity bit covers. The lecture then works the standard example r = 3, giving n = 7 and k = 4, i.e., the Hamming code C(7,4) with dmin = 3. Parity bits are placed at positions 1, 2, and 4; data bits occupy positions 3, 5, 6, and 7. Using even parity, the instructor specifies coverage sets: P1 checks positions 1, 3, 5, 7; P2 checks 2, 3, 6, 7; and P4 checks 4, 5, 6, 7. Handwritten binary sequences illustrate how each parity bit is computed from the covered positions.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The lecture opens with the slide titled “Hamming Codes,” stating that these codes are designed with dmin = 3, meaning they can detect up to two errors or correct one single error. The instructor emphasizes that the discussion focuses on the single-bit error-correcting code, a phrase underlined on screen. The formulas n = 2^r - 1 and k = n - r or k = 2^r - r - 1 are displayed, establishing the relationship among total bits n, data bits k, and redundancy r. A table begins mapping bit positions 1 through 20 to encoded data bits, with parity bits p1, p2, p4, p8, and p16 highlighted in green under “Parity bit coverage.”

  2. 2:00 – 5:00 02:00-05:00

    The instructor continues explaining the Hamming code structure, pointing to the slide with a pen and then gesturing while speaking. The same “Hamming Codes” slide remains visible, with the dmin = 3 bullet and the underlined focus on single-bit error correction. A hand-drawn diagram appears above the table showing two arrow segments labeled K and r, alongside a boxed formula reading k = 2^r - r - 1. The grid table maps bit positions 1 through 20 to alternating p and d labels, with X marks filling cells to show coverage by each parity bit. The instructor raises a pen to write on the board, beginning worked calculations.

  3. 5:00 – 10:00 05:00-10:00

    The lecture transitions to a concrete example. A new slide states, “For example, if r = 3, then n = 7 and k = 4. This is a Hamming code C(7, 4) with dmin = 3.” Handwritten annotations include k = 4 and a worked line y = 8 − 3 − 1, illustrating the formula application. The instructor explains that parity bits are placed at positions that are powers of two: 2^0 = 1, 2^1 = 2, and 2^2 = 4. A table maps positions 1 through 7 to P1, P2, D1, P4, D2, D3, and D4. Even-parity checks are listed: “For parity bit P1 we check position 1, 3, 5, 7,” “For parity bit P2 we check position 2, 3, 6, 7,” and “For parity bit P4 we check position 4, 5, 6, 7.” The instructor writes binary sequences on the right side of the screen and draws arrows from table positions to these sequences, demonstrating how each parity bit is calculated.

  4. 10:00 – 11:58 10:00-11:58

    The final segment reinforces the C(7,4) construction. The instructor displays the Hamming code parameters with n = 7 and k = 4, dmin = 3. The positions for parity bits are reiterated as powers of 2: 1, 2, and 4. The coverage sets for each parity bit are restated: P1 checks positions 1, 3, 5, and 7; P2 checks 2, 3, 6, and 7; P4 checks 4, 5, 6, and 7. A 7-bit table is drawn with data bits D1 through D4 and parity bits P1, P2, and P4. The instructor annotates the board with binary values and parity calculations, using a pointer to highlight specific positions in the table. The even-parity calculation process is illustrated with handwritten binary values, showing how the parity bits are determined from the covered data positions.

The lecture progresses systematically from general principles to a specific worked example. It begins by defining Hamming codes as error-correcting codes with dmin = 3, which provides the capability to detect two errors or correct one. The mathematical foundation is established through the formulas n = 2^r - 1 and k = 2^r - r - 1, linking the number of redundancy bits r to the total codeword length n and data bits k. A visual table maps bit positions to parity and data bits, with X marks illustrating the coverage pattern: each parity bit at position 2^i checks a specific subset of positions. The lecture then applies these principles to the canonical C(7,4) code with r = 3. Parity bits are placed at positions 1, 2, and 4 (powers of two), while data bits occupy the remaining positions. The instructor explicitly lists the coverage sets for each parity bit and demonstrates even-parity calculation using handwritten binary sequences. This structured approach—definition, formula, visual mapping, and worked example—provides a clear framework for understanding Hamming code construction.

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