feat(codes): add a QR Code encoder (versions 1-40, levels L-H) as a resumable job

Optimal numeric/alphanumeric/byte segmentation, Reed-Solomon over GF(256),
all eight masks scored like segno, steps sized for the 120,000-bytecode
watchdog; block tables generated from segno; a generic Matrix type with a
compact form for caching.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

1 files changed, 110 insertions, 0 deletions

diff --git a/source/codes/ReedSolomon.mc b/source/codes/ReedSolomon.mc
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+++ b/source/codes/ReedSolomon.mc
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+// Author:  cirakg & claude ai
+// Summary: Reed-Solomon error-correction codewords over GF(2^m) for QR (and later Data Matrix and Aztec).
+//
+// A Field is GF(2^m) with a primitive polynomial and the exponent the
+// generator's roots start at (QR: 0x11D, base 0; Data Matrix: 0x12D,
+// base 1). Generator polynomials are cached per field and length. ecc()
+// is the remainder of data * x^n divided by the generator: the standard
+// systematic encoding, one multiply-accumulate per data symbol and EC
+// symbol, so a caller budgets blocks per job step.
+
+import Toybox.Lang;
+
+module ReedSolomon {
+
+    class Field {
+        var size as Number;
+        var exp as Array<Number>;
+        var log as Array<Number>;
+        var base as Number;
+        private var _generators as Dictionary<Number, Array<Number>> = {} as Dictionary<Number, Array<Number>>;
+
+        // primitive: the field polynomial including x^m (0x11D for GF(256)).
+        function initialize(primitive as Number, fieldSize as Number, generatorBase as Number) {
+            size = fieldSize;
+            base = generatorBase;
+            exp = new [2 * fieldSize] as Array<Number>;
+            log = new [fieldSize] as Array<Number>;
+            var x = 1;
+            for (var i = 0; i < fieldSize - 1; i++) {
+                exp[i] = x;
+                log[x] = i;
+                x <<= 1;
+                if (x >= fieldSize) {
+                    x ^= primitive;
+                }
+            }
+            // Doubled table: exp[a + b] needs no modulo.
+            for (var i = fieldSize - 1; i < 2 * fieldSize; i++) {
+                exp[i] = exp[i - (fieldSize - 1)];
+            }
+            log[0] = 0;
+        }
+
+        function multiply(a as Number, b as Number) as Number {
+            if (a == 0 || b == 0) {
+                return 0;
+            }
+            return exp[log[a] + log[b]];
+        }
+
+        // Coefficients of the generator with n roots, highest degree first,
+        // without the leading 1.
+        function generator(n as Number) as Array<Number> {
+            var cached = _generators[n];
+            if (cached != null) {
+                return cached;
+            }
+            var g = [1] as Array<Number>;
+            for (var i = 0; i < n; i++) {
+                // Multiply by (x - a^(base + i)); in GF(2^m) minus is plus.
+                var root = exp[(base + i) % (size - 1)];
+                var next = new [g.size() + 1] as Array<Number>;
+                for (var k = 0; k < next.size(); k++) {
+                    next[k] = 0;
+                }
+                for (var k = 0; k < g.size(); k++) {
+                    next[k] = next[k] ^ g[k];
+                    next[k + 1] = next[k + 1] ^ multiply(g[k], root);
+                }
+                g = next;
+            }
+            var coefficients = g.slice(1, null) as Array<Number>;
+            _generators[n] = coefficients;
+            return coefficients;
+        }
+    }
+
+    // The n error-correction symbols of data (both as symbol values).
+    function ecc(field as Field, data as Array<Number>, n as Number) as Array<Number> {
+        var g = field.generator(n);
+        var rem = new [n] as Array<Number>;
+        for (var k = 0; k < n; k++) {
+            rem[k] = 0;
+        }
+        var exp = field.exp;
+        var log = field.log;
+        var logG = new [n] as Array<Number>;
+        for (var k = 0; k < n; k++) {
+            logG[k] = g[k] == 0 ? -1 : log[g[k]];
+        }
+        for (var i = 0; i < data.size(); i++) {
+            var factor = data[i] ^ rem[0];
+            // Shift the remainder left by one symbol.
+            for (var k = 0; k < n - 1; k++) {
+                rem[k] = rem[k + 1];
+            }
+            rem[n - 1] = 0;
+            if (factor != 0) {
+                var lf = log[factor];
+                for (var k = 0; k < n; k++) {
+                    var lg = logG[k];
+                    if (lg >= 0) {
+                        rem[k] = rem[k] ^ exp[lf + lg];
+                    }
+                }
+            }
+        }
+        return rem;
+    }
+}