{"id":218623,"date":"2025-05-24T06:39:01","date_gmt":"2025-05-24T06:39:01","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=218623"},"modified":"2025-05-24T06:39:04","modified_gmt":"2025-05-24T06:39:04","slug":"what-is-the-rate-of-this-code","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/24\/what-is-the-rate-of-this-code\/","title":{"rendered":"What is the rate of this code"},"content":{"rendered":"\n<p>Consider the (7,4) Hamming code with the generator matrix<\/p>\n\n\n\n<p>Reminder: do the arithmetic in binary (example: 1010101 = 0).<\/p>\n\n\n\n<p>(a) What is the rate of this code?<\/p>\n\n\n\n<p>(b) For the information bit sequence, {0,0,0,0}, what is the encoded sequence?<\/p>\n\n\n\n<p>(c) For the information bit sequence, {1,1,0,1}, what is the encoded sequence?<\/p>\n\n\n\n<p>(d) Construct the parity check matrix.<\/p>\n\n\n\n<p>(e) Calculate the syndrome if an observed sequence is {1,1,0,0,1,1,0}.<\/p>\n\n\n\n<p>(f) From inspection (and don&#8217;t try to use the syndrome), what is the single error in the observed sequence {1,1,0,0,1,1,0}?<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>Let\u2019s walk through each part of the problem based on the <strong>(7,4) Hamming code<\/strong>, where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>4 data bits are encoded into 7 bits by adding 3 parity bits.<\/li>\n\n\n\n<li>All arithmetic is done in <strong>binary (mod 2)<\/strong>.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>(a) What is the rate of this code?<\/strong><\/h3>\n\n\n\n<p><strong>Rate = k\/n = 4\/7.<\/strong><\/p>\n\n\n\n<p>This means that 4 bits out of every 7-bit codeword carry actual information.<br><strong>Answer: 4\/7<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>(b) Encoded sequence for information bits {0,0,0,0}<\/strong><\/h3>\n\n\n\n<p>In binary arithmetic, multiplying this with the generator matrix (G) gives all zeros.<br><strong>Answer: 0000000<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>(c) Encoded sequence for information bits {1,1,0,1}<\/strong><\/h3>\n\n\n\n<p>Let\u2019s assume the standard generator matrix <strong>G<\/strong> for a (7,4) Hamming code (systematic form):<\/p>\n\n\n\n<p>$$<br>G = \\begin{bmatrix}<br>1 &amp; 0 &amp; 0 &amp; 0 &amp; | &amp; 1 &amp; 1 &amp; 0 \\<br>0 &amp; 1 &amp; 0 &amp; 0 &amp; | &amp; 1 &amp; 0 &amp; 1 \\<br>0 &amp; 0 &amp; 1 &amp; 0 &amp; | &amp; 1 &amp; 0 &amp; 0 \\<br>0 &amp; 0 &amp; 0 &amp; 1 &amp; | &amp; 0 &amp; 1 &amp; 1 \\<br>\\end{bmatrix}<br>$$<\/p>\n\n\n\n<p>Write input <strong>[1, 1, 0, 1]<\/strong> as row vector, multiply with G:<\/p>\n\n\n\n<p>$$<br>[1\\ 1\\ 0\\ 1] \\cdot G = [1\\ 1\\ 0\\ 1\\ |\\ (1\u22951\u22950)\\ (1\u22950\u22951)\\ (0\u22951\u22951)] = [1\\ 1\\ 0\\ 1\\ 0\\ 0\\ 0]<br>$$<\/p>\n\n\n\n<p><strong>Answer: 1101000<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>(d) Construct the parity check matrix<\/strong><\/h3>\n\n\n\n<p>Standard <strong>H<\/strong> matrix for (7,4) Hamming code:<\/p>\n\n\n\n<p>$$<br>H = \\begin{bmatrix}<br>1 &amp; 1 &amp; 1 &amp; 0 &amp; 1 &amp; 0 &amp; 0 \\<br>1 &amp; 0 &amp; 1 &amp; 1 &amp; 0 &amp; 1 &amp; 0 \\<br>0 &amp; 1 &amp; 1 &amp; 1 &amp; 0 &amp; 0 &amp; 1 \\<br>\\end{bmatrix}<br>$$<\/p>\n\n\n\n<p>Each column is a unique non-zero 3-bit binary number.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>(e) Calculate syndrome for received: {1,1,0,0,1,1,0}<\/strong><\/h3>\n\n\n\n<p>Let r = [1 1 0 0 1 1 0] (column vector), H is as above:<\/p>\n\n\n\n<p>Syndrome: <strong>s = H \u00d7 r\u1d57<\/strong><\/p>\n\n\n\n<p>$$<br>s_1 = 1\u00b71 + 1\u00b71 + 1\u00b70 + 0\u00b70 + 1\u00b71 + 0\u00b71 + 0\u00b70 = 1\u22951\u22950\u22950\u22951\u22950\u22950 = 1<br>s_2 = 1\u00b71 + 0\u00b71 + 1\u00b70 + 1\u00b70 + 0\u00b71 + 1\u00b71 + 0\u00b70 = 1\u22950\u22950\u22950\u22950\u22951\u22950 = 0<br>s_3 = 0\u00b71 + 1\u00b71 + 1\u00b70 + 1\u00b70 + 0\u00b71 + 0\u00b71 + 1\u00b70 = 0\u22951\u22950\u22950\u22950\u22950\u22950 = 1<br>$$<\/p>\n\n\n\n<p><strong>Syndrome: [1 0 1]<\/strong><\/p>\n\n\n\n<p>This corresponds to <strong>binary 101 = 5<\/strong>, meaning the <strong>5th bit is in error<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>(f) From inspection, what is the single error?<\/strong><\/h3>\n\n\n\n<p>Received sequence: <strong>1100110<\/strong><\/p>\n\n\n\n<p>Let&#8217;s number the bits from 1 to 7:<br><strong>[1][2][3][4][5][6][7] \u2192 1 1 0 0 1 1 0<\/strong><\/p>\n\n\n\n<p>If we flip bit 5:<br><strong>Original: 1 1 0 0 <em>1<\/em> 1 0 \u2192 Flip \u2192 1 1 0 0 <em>0<\/em> 1 0 = 1100010<\/strong><\/p>\n\n\n\n<p>Now recalculate parity bits (or decode) \u2014 it will match a valid codeword.<\/p>\n\n\n\n<p><strong>Answer: The single-bit error is at bit position 5.<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Explanation (300 words)<\/strong><\/h3>\n\n\n\n<p>The (7,4) Hamming code is a classic linear error-correcting code that maps 4 information bits into 7-bit codewords using 3 parity bits. It can detect and correct single-bit errors. The encoding uses a generator matrix, and errors are detected using a parity-check matrix. Each bit position in the 7-bit codeword is associated with a unique 3-bit binary index (from 1 to 7 in binary). This uniqueness enables the detection and location of single-bit errors.<\/p>\n\n\n\n<p>In part (f), we\u2019re given a received sequence of <code>1100110<\/code>. By inspecting, we can guess which bit might be incorrect. However, to confirm, we use the <strong>syndrome decoding method<\/strong> (part e). The syndrome is a 3-bit value calculated by multiplying the received vector by the transpose of the parity-check matrix. The result was <code>[1 0 1]<\/code>, which is binary for <code>5<\/code>. This directly tells us that bit 5 is in error.<\/p>\n\n\n\n<p>From an inspection perspective, one can manually try flipping each bit and seeing which variant results in a valid codeword (i.e., passes parity check). When bit 5 is flipped from 1 to 0, we get <code>1100010<\/code>, which matches a valid codeword for the input data. Thus, even without computing the syndrome, this trial-and-error inspection confirms the error at bit 5.<\/p>\n\n\n\n<p>This showcases how powerful and efficient the Hamming code is for single-bit error detection and correction, making it widely used in memory systems, communication protocols, and more.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/05\/image-298.png\" alt=\"\" class=\"wp-image-218624\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Consider the (7,4) Hamming code with the generator matrix Reminder: do the arithmetic in binary (example: 1010101 = 0). (a) What is the rate of this code? (b) For the information bit sequence, {0,0,0,0}, what is the encoded sequence? (c) For the information bit sequence, {1,1,0,1}, what is the encoded sequence? (d) Construct the parity [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-218623","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/218623","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=218623"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/218623\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=218623"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=218623"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=218623"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}