Counting Irreducible Polynomials of Degree <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2811508pt" id="M1" height="8.47856pt" version="1.1" viewBox="-0.0657574 -8.19741 7.70668 8.47856" width="7.70668pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><path id="g113-115" d="M393 379C402 394 400 411 393 422C384 437 365 448 348 448C301 448 237 372 186 285H182L193 335C210 408 205 448 178 448C150 448 80 402 29 344L45 321C80 355 114 373 122 373C128 373 130 365 124 330C106 228 76 98 50 -5L57 -12C82 -5 112 3 132 6L172 203C196 256 234 304 254 329C275 355 293 367 306 367C318 367 330 360 342 348C347 343 355 343 365 350S386 367 393 379Z"/></g></svg> over <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-7.527801pt" id="M2" height="19.4769pt" version="1.1" viewBox="-0.0657574 -11.9491 20.8207 19.4769" width="20.8207pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><path id="g197-17" d="M567 618V662H70V0H236V304H501V348H236V618H567ZM192 44H114V618H192V44Z"/></g><g transform="matrix(.013,0,0,-0.013,8.387,4.308)"><path id="g50-114" d="M481 430L460 436C440 442 393 451 366 451C348 451 325 448 313 446C267 437 195 409 147 367C79 308 24 211 24 105C24 37 56 -12 93 -12C121 -12 156 1 199 32C246 66 313 122 349 168H351L286 -147C274 -205 261 -215 210 -224L199 -226L193 -255L438 -243L442 -214L417 -212C364 -208 359 -200 370 -140L433 204C455 324 469 389 481 430ZM391 385C382 336 366 260 358 238C321 178 209 59 143 59C129 59 115 83 115 131C115 207 152 317 220 373C243 392 282 403 314 403C348 403 374 395 391 385Z"/></g><g transform="matrix(.009,0,0,-0.009,14.635,-.09)"><path id="g176-111" d="M513 91L497 125C461 87 438 71 424 71C416 71 410 77 419 106C440 174 459 243 472 310C497 441 467 454 445 454C419 454 393 443 360 424C319 400 237 344 180 265H178L196 336C217 421 207 454 182 454C146 454 84 416 25 354L42 318C66 342 103 368 113 368C119 368 124 362 117 333L31 -2L41 -12C59 -3 91 5 119 11C133 76 146 123 162 180C214 262 337 384 383 384C404 384 411 371 391 304C370 233 350 165 333 90C318 24 328 -12 359 -12C385 -12 449 23 513 91Z"/></g></svg> and Generating Goppa Codes Using the Lattice of Subfields of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-7.527801pt" id="M3" height="19.4769pt" version="1.1" viewBox="-0.0657574 -11.9491 24.689 19.4769" width="24.689pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><use xlink:href="#g197-17"/></g><g transform="matrix(.013,0,0,-0.013,8.387,4.308)"><use xlink:href="#g50-114"/></g><g transform="matrix(.009,0,0,-0.009,14.635,-.09)"><use xlink:href="#g176-111"/></g><g transform="matrix(.009,0,0,-0.009,19.302,-.09)"><path id="g176-115" d="M401 382C409 397 408 412 399 428C392 441 374 454 353 454C302 454 241 373 198 302H196L204 340C218 406 209 454 182 454C154 454 85 408 31 347L50 314C93 354 121 372 129 372C139 372 140 360 131 317L59 -5L64 -12C91 -5 123 3 147 4C162 97 174 169 186 211C205 255 248 312 265 334S300 366 307 366C325 366 337 361 353 345C356 342 368 342 374 347C383 354 392 366 401 382Z"/></g></svg>

<table>Lattice of subfields of <svg height="12.3659pt" id="M190" style="vertical-align:-3.31356pt" version="1.1" viewbox="-0.0498162 -9.05234 18.8276 12.3659" width="18.8276pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><path d="M567 618V662H70V0H236V304H501V348H236V618H567ZM192 44H114V618H192V44Z" id="g197-17"></path><glyph.data ascent="693" descent="-218" horiz-adv-x="469" vert-adv-y="469"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,6.354,3.264)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.0068,0,0,-0.0068,10.973,-.068)"><path d="M436 222C436 297 385 359 295 376V378C320 391 343 407 363 428S394 474 394 501C394 578 333 635 239 635C186 635 138 610 110 577L63 514L91 490C114 525 158 571 211 571C266 571 306 541 306 483C306 411 234 371 141 343L151 306C167 311 192 317 219 317C276 317 344 282 344 194C346 102 296 46 227 46C166 46 125 76 99 97C91 104 81 103 71 97C62 90 47 75 46 62C44 50 48 39 62 26C77 10 120 -12 168 -12C242 -12 436 62 436 222Z" id="g176-52"></path><glyph.data ascent="3439" descent="-2858" horiz-adv-x="494" vert-adv-y="494"></glyph.data></g><g transform="matrix(.0068,0,0,-0.0068,14.319,-.068)"><path d="M455 312C455 472 400 635 248 635C95 635 39 458 39 312C39 144 93 -12 248 -12C400 -12 455 166 455 312ZM361 314C361 173 337 30 249 30C163 30 131 174 131 313C131 451 155 594 248 594C334 594 361 454 361 314Z" id="g176-49"></path><glyph.data ascent="3439" descent="-2858" horiz-adv-x="494" vert-adv-y="494"></glyph.data></g></svg>.</table>

Journal of Discrete Mathematics

fig2

Figure 2

Figure 2: Counting Irreducible Polynomials of Degree <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2811508pt" id="M1" height="8.47856pt" version="1.1" viewBox="-0.0657574 -8.19741 7.70668 8.47856" width="7.70668pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><path id="g113-115" d="M393 379C402 394 400 411 393 422C384 437 365 448 348 448C301 448 237 372 186 285H182L193 335C210 408 205 448 178 448C150 448 80 402 29 344L45 321C80 355 114 373 122 373C128 373 130 365 124 330C106 228 76 98 50 -5L57 -12C82 -5 112 3 132 6L172 203C196 256 234 304 254 329C275 355 293 367 306 367C318 367 330 360 342 348C347 343 355 343 365 350S386 367 393 379Z"/></g></svg> over <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-7.527801pt" id="M2" height="19.4769pt" version="1.1" viewBox="-0.0657574 -11.9491 20.8207 19.4769" width="20.8207pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><path id="g197-17" d="M567 618V662H70V0H236V304H501V348H236V618H567ZM192 44H114V618H192V44Z"/></g><g transform="matrix(.013,0,0,-0.013,8.387,4.308)"><path id="g50-114" d="M481 430L460 436C440 442 393 451 366 451C348 451 325 448 313 446C267 437 195 409 147 367C79 308 24 211 24 105C24 37 56 -12 93 -12C121 -12 156 1 199 32C246 66 313 122 349 168H351L286 -147C274 -205 261 -215 210 -224L199 -226L193 -255L438 -243L442 -214L417 -212C364 -208 359 -200 370 -140L433 204C455 324 469 389 481 430ZM391 385C382 336 366 260 358 238C321 178 209 59 143 59C129 59 115 83 115 131C115 207 152 317 220 373C243 392 282 403 314 403C348 403 374 395 391 385Z"/></g><g transform="matrix(.009,0,0,-0.009,14.635,-.09)"><path id="g176-111" d="M513 91L497 125C461 87 438 71 424 71C416 71 410 77 419 106C440 174 459 243 472 310C497 441 467 454 445 454C419 454 393 443 360 424C319 400 237 344 180 265H178L196 336C217 421 207 454 182 454C146 454 84 416 25 354L42 318C66 342 103 368 113 368C119 368 124 362 117 333L31 -2L41 -12C59 -3 91 5 119 11C133 76 146 123 162 180C214 262 337 384 383 384C404 384 411 371 391 304C370 233 350 165 333 90C318 24 328 -12 359 -12C385 -12 449 23 513 91Z"/></g></svg> and Generating Goppa Codes Using the Lattice of Subfields of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-7.527801pt" id="M3" height="19.4769pt" version="1.1" viewBox="-0.0657574 -11.9491 24.689 19.4769" width="24.689pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><use xlink:href="#g197-17"/></g><g transform="matrix(.013,0,0,-0.013,8.387,4.308)"><use xlink:href="#g50-114"/></g><g transform="matrix(.009,0,0,-0.009,14.635,-.09)"><use xlink:href="#g176-111"/></g><g transform="matrix(.009,0,0,-0.009,19.302,-.09)"><path id="g176-115" d="M401 382C409 397 408 412 399 428C392 441 374 454 353 454C302 454 241 373 198 302H196L204 340C218 406 209 454 182 454C154 454 85 408 31 347L50 314C93 354 121 372 129 372C139 372 140 360 131 317L59 -5L64 -12C91 -5 123 3 147 4C162 97 174 169 186 211C205 255 248 312 265 334S300 366 307 366C325 366 337 361 353 345C356 342 368 342 374 347C383 354 392 366 401 382Z"/></g></svg> 

Journal of Discrete Mathematics

Counting Irreducible Polynomials of Degree over and Generating Goppa Codes Using the Lattice of Subfields of

Figure 2